Ch. 3 Properties of Numbers
Remainders after division by 10^n
When a whole number is divided by 10, the remainder will be the units digit of the dividend (numerator). When a whole number is divided by 100, the remainder will be the last two digits of the dividend, etc. For example, when 153 is divided by 10, the result is 15.3, which is 15 + 3/10(remainder3).
The prime factorization of a perfect square will contain only even exponents
A perfect square, other than 0 and 1, is a number such that all of its prime factors have even exponents. For example, the prime factorization of the perfect square 64 is 2^6. Notice that 6 is an even exponent. In addition, consider the perfect square 100. The prime factorization of 100 is 10^2 = 52 × 22. Again, notice that all of the exponents in the prime factorized form are even. Example: If z is a positive integer, and 420z is a perfect square, what is the lowest possible value of z? Essentially, we need to find the prime factorization of the (420) portion of the question and determine which factors have even exponents and which ones do not. Since perfect squares can only contain even exponents, we need to find the prime factors with odd exponents and find out what needs to be multiplied in to make those even exponents (multiplying one for each). Go to example problem 62 on 3.18.1 for a good illustration of this. Also note: sometimes the gmat will present a perfect square as x^2 or say its equivalent to an integer squared. Remember that these are all ways to present perfect squares.
Number of prime factors vs. number of unique prime factors
On the GMAT, you may encounter problems that discuss the total number of prime factors of a particular number and the total number of unique prime factors of a particular number. Although these two phrases may appear similar, there are some subtle but important differences between them. We can determine the total number of prime factors of a number by counting the individual prime factors in the prime factorization of that number.
Prime numbers
A prime number is any integer greater than 1 that has no factors other than 1 and itself. Prime numbers have only two factors: 1 and themselves. Therefore, they are only divisible by two numbers. The number 2 is the only even prime number, and it is also the smallest prime number. Memorize all 25 prime numbers under 100.
Terminating decimals
A terminating decimal is one that has a finite number of non-zero digits to the right of the decimal point. ⇒ 1/10 = 0.10 or ⇒ 1/25 = 0.04 Some decimals do not terminate. These decimals have an infinite number of non-zero digits to the right of the decimal point. ⇒ 1/6 = 0.1666666666... The decimal equivalent of a fraction will terminate if and only if the denominator of the reduced fraction has a prime factorization that contains only 2s or 5s, or both. If the prime factorization of the reduced fraction's denominator contains anything other than 2s or 5s, the decimal equivalent will not terminate.
Even and odd numbers
All even numbers have an even units digit, and all odd numbers have an odd units digit. Even integers can be represented by 2n, where n is an integer, and odd integers can be represented by either 2n - 1, or by 2n + 1, where n is an integer. For example, if n = 0, then 2n = 2(0) = 0, which is an even number. Also, 2n - 1 = 0 - 1 = -1, which is an odd number, and 2n + 1 = 0 + 1 = 1, which is also an odd number.
Adding And Subtracting Remainders
Remainders can be added, but we must correct for any excess remainders at the end of the addition. Similarly, remainders can be subtracted, but we must correct for negative remainders. If the result is a negative remainder with a positive quotient, take 1 away from the quotient and add a whole number to the fraction. (check?)
Word problems involving divisibility
Remember to rewrite and simplify word problem equations whenever possible.
Even division (qualities of factors and multiples)
Remember, a factor of a positive integer n is a positive integer that divides into n, while a multiple of n is the product of such a factor and some positive integer. Given two positive integers, x and y, xy will yield an integer if x is a multiple of y (or, alternatively, if y is a factor of x). ⇒ 5 divides into 100; thus, 5 is a factor of 100 (and 100 is a multiple of 5, since 5 × 20 = 100). There are a number of phrases that should alert us that dividing some number x by some number y will result in an integer: ⇒"y is a factor of x" ⇒"y is a divisor of x" ⇒"y divides into x (evenly)" ⇒"x is a multiple of y" ⇒"x is a dividend of y" ⇒"x is divisible by y"
Multiplication rules for even and odd numbers
The product of an even number and any integer is always even. In a multiplication of a set of two or more integers, if any of these numbers is even, the product will be even. The product of an odd number and any odd number is always odd. In a multiplication of a set of two or more integers, if every number is odd, the product will be odd.
Perfect squares
The square root of a perfect square will always be an integer. Thus, in taking the square root of a nonnegative integer x, if the result is an integer, then x is a perfect square. 0√=0, 1√=1, 4√=2, 9√=3, 16√=4, 25√=5, 36√=6, 49√=7 Reminder: All perfect squares must end in 0, 1, 4, 5, 6, or 9. A number that ends in 2, 3, 7, or 8 can never be a perfect square.
Factorial notation and the division properties of factorials
n! represents the multiplication of all of the integers from 1 to n, inclusive, if n ≥ 1. The product of any set of consecutive integers is divisible by any of the integers in the set. In addition, the product is divisible by any of the factor combinations of these numbers. For example, the product 5 × 6 × 7 × 8 is 1,680, which is divisible by 5, 6, 7 and 8. It's also divisible by the product of any of factor combinations of 5, 6, 7 and 8, such as 5 × 6 = 30, 7 × 8 = 56 and 5 × 6 × 8 = 240. n! must be divisible by all of the integers from 1 to n inclusive. In addition, n! must be divisible by any factor combinations of 1 to n inclusive.
Leading zeros in decimals (not a perfect power of 10)
"Leading zeros" are any zeros that appear after the decimal point and before the first nonzero digit. For example, 0.002 has 2 leading zeros, and 0.7310 has no leading zeros. When the denominator X is not a perfect power of 10 If X is an integer with k digits, and if X is not a perfect power of 10, then 1/X will have k - 1 leading zeros. 1/8 = 0.125, no leading zeroes 1/48 = 0.0208, one leading zero 1/200 = 0.005, two leading zeros Remember to check that your denominator formulas do not simplify to a perfect power of 10. This may not be obvious at first (i.e. 5 x 2 pairs, which = 10) Be sure to check if the question is asking for trailing zeroes versus number of digits!
Factors
For any positive integers x and y, y is a factor of x if and only if xy is an integer. Furthermore, 1 ≤ y ≤ x. 1, 2, 3, 6, 9, and 18 are factors of 18 because these numbers all divide into 18. In word problems, multiples go in the numerator and factors go in the denominator, such that multiple/factor.
The LCM and GCF when one number divides evenly into the other
If we know that positive integer y divides evenly into positive integer x, the LCM of x and y is x and the GCF of x and y is y. For example, since 25 divides evenly into 100 then LCM(100, 25) = 100 and GCF(100, 25) = 25.
Using trailing zeros to determine the number of digits in an integer
So, when given a number such as 25^10 × 8^6, to determine the digits in a number, use the following steps: Step 1: Prime factorize the number. Step 2: Count the number of (5 × 2) pairs. Each (5 × 2) pair contributes one trailing zero. Step 3: Collect the number of unpaired 5s or 2s, along with other nonzero prime factors (if any) and multiply them together. Count the number of digits in this product. Step 4: Sum the number of digits from steps 2 and 3. Be sure to check if the question is asking for trailing zeroes versus number of digits!
The factor chart: finding the factors of a number
Start by identifying the two most obvious factors of a given number: 1 and the number itself (n). Then, begin asking what whole integer results as you divide up from 1..2..3..4..5... from n. This is a good process to build out a factor chart.
The range of possible remainders
A remainder must be a non-negative integer that is less than the divisor. That is, if a divisor is n, then possible remainders could be any integers ranging from 0 to (n - 1), inclusive. For example, if the divisor is 5 (n = 5), the possible remainders are 0, 1, 2, 3, and 4.
Integers
An integer is a number that can be written without a decimal or a fractional component. Whole numbers are nonnegative integers. That is, the positive integers and the number zero form the set of whole numbers. For example, -100, -20, -5, -1, 0, 1, 5, 20, and 100 are all integers. Numbers such as 1.2, -3/4, and π, which is approximately 3.14, are not integers Zero is neither positive nor negative and has no sign associated with it. Whole numbers are specifically nonnegative integers.
Number patterns - divisors and powers
As we divide positive integers by a constant divisor, the remainders that result will exhibit a pattern. Just as all divisors have a remainder pattern, when we divide a certain divisor into powers of a certain base, a similar pattern will result. When a positive integer is raised to consecutive powers that are positive integers (that is, if both the base and the exponent are positive integers), a pattern will arise in the units digit of the product. The Gmat will give you large and tedious equations to solve. If an answer differentiator is the remainder of a mixed fraction (or can be converted) then you should employ this trick. Rather than memorize all of the number patters, plug in values starting with 1 until you recognize a pattern. The pattern will end when it begins to repeat itself. Is the pattern consistent? Only when even or odd? In some kind of other order? Additionally, for base numbers greater than 9, you can still follow the rule above by only considering the units digit of the base number. Example: 12^2 = 144 and 2^= 4 Example: what is the units digit of 533431^9? Since the base number ends in a 1, we know that the answer must end in a 1 as well.
The prime factorization of a perfect cube will contain only exponents that are multiples of 3
If some number x (that is not 0 or 1) is a perfect cube, the prime factorization of x must contain only exponents that are multiples of 3. For example, the prime factorization of the perfect cube 64 is 4^3 = 2^6. Notice that 6 is a multiple of 3. Also consider the perfect cube 125. The prime factorization of 125 is 5^3. Again, notice that the exponent is a multiple of 3. Similar to the perfect squares prime factorization, we need to find what value is needed to multiply the number given in order to get cubed prime factors in the result. Go to example problem 65 on 3.19.1 for a good illustration of this.
The number of unique prime factors in a number does not change when that number is raised to a positive integer exponent
If some number x has y unique prime factors, then x^n (where n is a positive integer) will have the same y unique prime factors. For example, the prime factorization of 25 is 52. That is, 25 has one unique prime factor, 5. If we square 25 we produce 625, which has the prime factorization of 54. Notice that although 625 contains more 5s than 25, the only prime factor that 625 has is 5, which is the same prime factor that 25 has.
LCM and GCF Properties
If the LCM of x and y is p and the GCF of x and y is q, then xy = pq. That is, xy = LCM(x, y) × GCF(x, y). Again, you can have more than two numbers in a set for this formula to work.
Leading zeros in decimals (is a perfect power of 10)
If the denominator is a perfect power of 10 (i.e. 10, 100, 1000...) then: If X is an integer with k digits, and X is a perfect power of 10, then 1/X will have (k - 2) leading zeros in its decimal form. Be sure to check if the question is asking for trailing zeroes versus number of digits!
Factors of factors
If x and y are positive integers and xy is an integer, then x/(any factor of y) is also an integer. That is, if y is a factor of x, and z is a factor of y, then z is a factor of x.
Divisibility with exponents
If x, a, and b are integers, and x ≠ 0, in order for x^a/x^b to be an integer, a≥b. If b > a, a proper fraction will result. That is, x^a is divisible by x^b if and only if a ≥ b.
If z is divisible by both x and y, z must also be divisible by the LCM of x and y
If z is divisible by both x and y, z must also be divisible by the LCM of x and y. Note: This does not imply anything by any number greater than the LCM. We can only say that Z is divisible by the prime factors of X and Y up to the LCM and no further.
Evenly spaced sets
In an evenly spaced set, the numbers in the set increase by the same amount and therefore share a common difference. For example, {11, 22, 33, 44, 55, 66} and {1, 2, 3, 4, 5, 6} are evenly spaced sets, since the numbers within each set are increasing by the same amount. In the first set, the numbers are increasing by 11, and in the second set the numbers are increasing by 1.
Prime factorization tree
In order to start the tree, we must find one number that divides evenly into your original number. Let's let n=192. Keep breaking down multiples until you end up with only prime numbers left. Multiple each of the prime numbers together and the product will be the original number 192. 2 × 2 × 2 × 2 × 2 × 2 × 3 = (2^6)(3^1) = 192. So we can say there are six factors of 2, and one factor of 3.
Determining if a number has a zero in the units digit
In whole numbers, trailing zeros are created by (5 × 2) pairs. Each (5 × 2) pair creates one trailing zero. Thus, the number of trailing zeros of a number is the number of (5 × 2) pairs in the prime factorization of that number. We see that 30 = 6 × 5 = 3 × 2 × 5. Notice that there is one (5 × 2) pair, and thus 30 has a units digit of zero. Additionally, each factor of 10 creates a trailing zero. Because the number 10 can be prime factorized into 5 × 2, we can restate this rule to say that each (5 × 2) pair creates a trailing zero. Additionally, any factorial ≥5! will always have zero as its units digit. This is because they will always contain a (5 x 2) pair.
Number of Trailing Zeros
It could be the case that in a number we have more factors of 5 than factors of 2, or vice versa. In such a case, the limiting factor is the number of 5s and 2s that are present, whichever is fewer. For example, how many trailing zeros are in the number 5^300 × 2^298? Since there are 298 2s, we can match those 298 2s with 298 of the 300 5s to create a number with 298 trailing zeros. Notice: Two lone 5s remain, but these do not affect the number of trailing zeros in the number. Be sure to check if the question is asking for trailing zeroes versus number of digits!
Properties of one
One is a factor of all numbers, and all numbers are multiples of 1. One is the only number with exactly 1 factor. One is an odd number. One is not a prime number. (The first prime number is 2.)
Converting a decimal remainder to an integer
One last trick worth memorizing is one that converts a decimal remainder into an exact integer remainder. For example, 9/5=1.8. To determine what the integer remainder is, we can multiply the decimal component of the result of the division (0.8) by the divisor (5) to get 0.8 x 5 = 4. Thus the remainder is 4.
Finding the LCM
Step 1: Find the prime factorization of each integer. That is, prime factorize each integer and put the prime factors of each integer in exponent form. Step 2: Of any repeated prime factors among the integers in the set, take only those with the largest exponent. For example, if we had 3^2 and 3^3, we'd choose 3^3 and not 3^2. If we're left with two of the same power (for example, 3^2 and 3^2), just take that number once. Step 3: Of what is left, take all non-repeated prime factors of the integers. Step 4: Multiply together what you found in Steps 2 and 3. The result is the least common multiple. Note: This same rule applies even if there are more than two numbers in a set.
Using the LCM to solve repeating pattern questions
The LCM can be used to determine when two processes that occur at differing rates or times will coincide. For example, let's say that blinking light L flashes once every 32 seconds, and blinking light M flashes once every 12 seconds. If both lights flash together at 8:00:00 PM, when will be the next time the lights will flash together again? The two lights will next flash together at the LCM of 32 seconds and 12 seconds. 32 = 2 × 2 × 2 × 2 × 2 = 2^5, and 12 = 2 × 2 × 3 = 2^2 × 3. Thus, the LCM of 32 and 12 is 2^5 × 3 = 96 seconds. Therefore, the lights will flash together again 96 seconds later at 8:01:36 PM.
Perfect cubes
The cube root of a perfect cube will always be an integer. Thus, when we take the cube root of a non-negative integer x, if the result is an integer, then x is a perfect cube. The first nine non-negative perfect cubes are 0, 1, 8, 27, 64, 125, 216, 343, and 512. 0√3=0, 1√3=1, 8√3=2, 27√3=3, 64√3=4, 125√3=5, 216√3=6, 343√3=7, 512√3=8
The greatest common factor (GCF)
The greatest common factor, or GCF, of a set of positive integers is the largest number that will divide into all of the numbers in the set. This is not trying to find a prime, but trying to find the largest factor shared across each number in the set. Step 1: Find the prime factorization of each number. That is, prime factorize each number and put the prime factors of each number in exponent form. Step 2: Identify repeated prime factors that exist in each of the numbers. Remember, each number in a set must contain one of these prime factors. Step 3: Of any repeated prime factors among the numbers, take only those with the smallest exponent. Step 4: Multiply together the numbers that you found in step 3; this product is the GCF. If a set of positive integers has no prime factors in common, the GCF of that set is 1. In addition, for any set of positive integers, the LCM will always be greater than or equal to the largest number in the set, and the GCF will always be less than or equal to the smallest number in the set. Note: If no repeated prime factors are found, the GCF is 1.
The least common multiple (LCM)
The least common multiple (LCM) of a set of positive integers is defined as the smallest positive integer into which all of the numbers in the set will divide. Put another way, the LCM is the smallest positive integer that is a multiple of all the integers in the set. For example, given the integers 2 and 5, the LCM is 10, since 10 is the smallest positive integer into which both 2 and 5 will divide. That is, 10 is the smallest positive multiple of both 2 and 5.
Divisibility rules
The rules of divisibility can help us to quickly determine whether two (or more) numbers are divisible by each other. Here are the divisibility rules for zero through 12: 0. No number is divisible by 0. 1. All numbers are divisible by 1. 2. A number is divisible by 2 if the ones digit is 0, 2, 4, 6, or 8 - that is, if the units digit is even. 3. A number is divisible by 3 if the sum of all the digits is divisible by 3. For example, 472,071 is divisible by 3 because the sum of its digits (4 + 7 + 2 + 0 + 7 + 1 = 21) is divisible by 3. 4. If the last two digits of a number are divisible by 4, then the number is divisible by 4. For example, the last two digits of 244 are 44, which is divisible by 4. Additionally, numbers ending in "00" is also divisible by 4. 5. A number is divisible by 5 if the last (ones) digit is a 0 or 5. For example, the numbers 55 and 70 are divisible by 5. 6. A number is divisible by 6 if the number in question is an even number whose digits sum to a multiple of 3 (and therefore is divisible by both 2 and 3, the factors of 6). 7. There is no rule for 7. 8. If the number is even, divide the last three digits by 8. If there is no remainder, then the original number is divisible by 8. 9. A number is divisible by 9 if the sum of all the digits is divisible by 9. For example, 479,655 is divisible by 9 because the sum of the digits (4 + 7 + 9 + 6 + 5 + 5 = 36) is divisible by 9. 10. If the ones digit is a zero, then the number is divisible by 10. 11. A number is divisible by 11 if the sum of the odd-numbered place digits minus the sum of the even-numbered place digits is divisible by 11. The odd-numbered place digits are the 1st, 3rd, 5th (and so on) digits on the left of the decimal point. Hence, they are the ones, hundreds, ten-thousands (and so on) digits. Similarly, the even-numbered place digits are the 2nd, 4th, 6th(and so on) digits on the left of the decimal point. Hence, they are the tens, thousands, hundred-thousands (and so on) digits. Let's have some examples. 253 is divisible by 11 because (2 + 3) - 5 = 0, which is divisible by 11 (remember 0 is divisible by any number except itself). 2,915 is divisible by 11 because (9 + 5) - (2 + 1) = 11, which is divisible by 11. 12. If a number is divisible by both 3 and 4, it is also divisible by 12.
Consecutive integers
There are three ways that evenly spaced sets commonly appear on the GMAT: 1) A set of consecutive integers (including sets of even and odd consecutive integers). 2) A set of consecutive multiples of a given number, such as {2, 4, 6, 8}, {10, 20, 30, 40, 50}, or {33, 66, 99, 132}. 3) A set of consecutive numbers with a given remainder when divided by some integer, such as {1, 6, 11, 16, 21}, or {3, 7, 11, 15, 19}. Note: Two consecutive integers will never share the same prime factors. Thus, the GCF of two consecutive integers is 1. That is, GCF(n, n+1) = 1. An important implication of this is that the greatest common factor (GCF) of two consecutive integers is 1.
Shortcut for Determining the Number of Primes in a Factorial When the Base of the Divisor Is Not a Prime Number
To determine the (largest) number of a non-prime number x that divides into y!, we perform the following steps: Step 1: Break x into prime factors. Step 2: Using the largest prime factor of x, apply the factorial divisibility shortcut to determine the quantity of that prime factor. The quantity determined represents the largest number of x that divides into y!. Essentially, you are applying the same shortcut as if it were a prime number, but only to the largest prime factor of the denominator. This is because you are basically trying to find factor pairs which can divide into your numerator, and as such you will be limited by your smallest number which would correlate to the limit placed on the largest of the two pairs.
Shortcut for Determining the Number of Primes in a Factorial When the Base of the Divisor Is a Power of a Prime Number
To determine the (largest) value of a non-prime number x (where x = pk, p is a prime, and k is an integer greater than 1) that divides into y!, we perform the following steps: Step 1: Express x = pk. Step 2: Apply the factorial divisibility shortcut to determine the quantity of p in y!. Then create and simplify an inequality to determine the largest number of x that divides into y!.
Finding the Number of Factors of a Particular Number
To determine the total number of positive integer factors of a number, we can take the following steps: Step 1: Find the prime factorization of the number. At the end of the process, make all exponents visible, including exponents of 1. Step 2:Add 1 to the value of each exponent, and then multiply these exponenets. The product will be the total number of factors of that number.
Multiplying Remainders
We can multiply remainders, but we must correct any excess remainders at the end of the multiplication. An excess remainder is a remainder greater than its divisor. We can correct an excess remainder by removing as many multiples of the original divisor as we can from the excess remainder. For example, let's say we wanted to know what the remainder was when we divided the product of 12 × 13 × 17 by 5. We can begin by dividing all of the numerators by 5 individually. Then multiply each of their remainders by each other and set it over the denominator. Remove excess if the result in an improper fraction so the end result is a mixed number. The final remainder will be the remainder of the original equation.
Converting A Remainder From Decimal Form To Fraction Form
We can only convert remainder fractions into decimals. The same cannot be said about the opposite. Converting 0.48 could be 48/100, but 48 is not necessarily the remainder that the problem is asking for. There are an infinite number of remainders this could be. When the division of two integers, x and y, results in a decimal, such as x/y=6.64, we can't be sure what the actual remainder is. Without knowing the value of x or y, or both, an infinite number of remainders is possible. What we can do is to determine what the most reduced fraction could be. The actual remainder must be a multiple of this number. Keep simplifying the fraction until its the smallest number possible. The actual remainder must be a multiple of that number. When a division of two integers results in a decimal, such as x/y=9.48, the actual remainder must be a multiple of the most reduced fractional remainder. That is, depending on the values of x and y, 0.48 can be expressed as a fraction in an infinite number of ways, but if we can determine the most reduced fractional value for 0.48, which is 12/25, we know that the actual remainder is a multiple of 12.
Shortcut for Determining the Number of Primes in a Factorial
What is the largest possible integer value of n such that 21!/3^n is an integer? To determine the (largest) number of a prime number x that divides into y!, we perform the following steps: Step 1: Divide y by x^1, then x^2, then x^3, etc. Keep track of the quotients while ignoring any remainders. We can stop dividing when y^xk produces a quotient of zero. Step 2: Add the quotients from the previous divisions; that sum represents the number of prime number x in the prime factorization of y!. Essentially, keep dividing the base of the numerator by the denominator, each time consecutively increasing by a power of 1 each time, until you create a number less than 1. Then sum up each step of the way to get the largest possible integer value. Alternatively, the first division resultant is considered the smallest possible.
Even and Odd Exponents Versus Positive and Negative Answers
When a nonzero base is raised to an even exponent, the result will always be positive. When a nonzero base is raised to an odd exponent, if the original base is positive, then the result will be positive. If the original base is negative, then the result will be negative. If x^4 > 0, what is the sign of x? (Unknown) If x^3 > 0, what is the sign of x? (Positive) If x^7 < 0, what is the sign of x? (Negative)
Addition and subtraction rules for even and odd numbers
When adding or subtracting two numbers, if both numbers are even, or both numbers are odd, the result will be even. If both numbers are some combination of even/odd, then the result will be odd.
Division rules for even and odd numbers
When an even number is divisible by an odd number, the result is an even number, and when an odd number is divisible by an odd number, the result is an odd number. That is even/odd is even and odd/odd is odd.
Remainders After Division By 5
When integers with the same units digit are divided by 5, the remainder is constant. 14/5 will yield the same remainder (also 4) as any numerator whose unit digit is also 4.
Multiplication and division with numbers of the same or different signs
When multiplying or dividing two numbers with the same signs, the product or quotient will always be positive. Example: -8/-2 = +4 , 12/4 = +3 When multiplying or dividing two numbers with different signs, the product or quotient will always be negative. Example: 8/-2 = -4 , -12/4 = -3
Properties of zero
Zero divided by any number other than zero is zero. Any number divided by zero is undefined (the quotient is not zero). The square root of zero is zero. Zero is the only number that is neither positive nor negative. Zero is a multiple of all numbers. Zero is not a factor of any number except itself. Any number raised to the zero power is equal to 1 (except zero) Zero is an even number.
