GRE Math: Number Properties
What is -4^2?
-(4^2)= 16. The answer is NOT (-4)^2= 16, because according to PEMDAS, exponents take precedence over subtraction.
How can you solve an equation with an unknown exponent? Ex. 2^x= 8.
-First, rewrite the value on the other side of the equal sign as an exponential expression with the same base as the other side: 2^x= 2^3 -Now that the bases are the same, you can ignore them and set the exponents equal to each other: x= 3
When deciding a problem asks about number properties, you can replace variables with E for even and O for odd instead of picking random numbers to better see relationships.
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What number would be exactly halfway between 5 and -5 on a number line?
0 would be the halfway point between -5 and 5. In general, if two numbers are opposite one another (pos./neg.), then they have the same absolute value and 0 is halfway between them.
What is x^0 equal to?
1
How can you simplify exponential expressions linked by multiplication/division?
1. Adding/subtracting exponents when expressions are being multiplied/divided. 2. Rewrite expressions that share a common exponent. Ex. 12^7/3^7= (12/3)^7= 4^7
What 2 things can absolute value brackets do to an expression?
1. Leave the expression the unchanged. This occurs whenever to expression is 0 or positive (greater than or equal to 0). 2. Change the sign of the whole expression. This occurs whenever the expression is 0 or negative (less than or equal to 0).
When can you simplify (reduce to one term) exponential expressions?
1. You can only simplify exponential expressions that are linked by multiplication/division, NOT by addition/subtraction (in some cases you CAN factor or otherwise manipulate expressions linked by addition/subtraction). 2. You can simplify exponential expressions linked by multiplication/division of they have either a base or an exponent in common.
What 3 parameters must be known for an evenly spaced sequence to be fully defined?
1. the smallest (first) or largest (last) number in the sequence 2. the increment (always 1 for consecutive integers) 3. the number of items in the sequence
What can 2^-3 be rewritten as?
1/2^3
What is the reciprocal of 2^-5?
1/2^5 = 1/32
How can 1/4 be expressed as an exponent?
1/4= 2^-2 (reciprocal role of negative exponents)
What is 10^6/2^6 simplified?
10^6/2^6= (10/2)^6= 5^6
What is 19 × 19^19 equal to?
19^20
What is the only even prime number?
2
What can (1/2)^y be rewritten as?
2^-y (reciporacal rule of negative exponents)
What is 2^4 × 3^4 simplified?
2^4 × 3^4= (2×3)^4= 6^4
How would you solve for the value of a factorial of a number? Ex. What is 4! × 6!
4! × 6! = (4×3×2×1) × (6×5×4×3×2×1) = 24 × 720 = 17,280
How many even integers are there between 12 and 24?
7. Apply the formula for counting consecutive multiples (even numbers= multiples of 2): 24 - 12= 12 ÷ 2 = 6 + 1 = 7
How many integers are there from 14 to 765, INCLUSIVE?
752. When dealing with counting terms in a long sequence OF CONSECUTIVE INTEGERS in which both extremes (smallest and largest items) should be counted, subtract the smallest number from the largest number and then "add one before you're done". 765-14= 751+1= 752
What is the quickest and easiest way to identify all of the factor pairs of a number when dealing with questions about divisibility?
A factor pair table provides a reliable way to make sure you find every factor of a number. Make one column for small factors of each pair and another for large factors if each pair and stop finding factors when the two begin to overlap.
What does it mean when you see a negative exponent in a problem?
A negative exponent means "put the term containing the exponent in the denominator of the fraction, and make the exponent positive". DIVIDE by the base a certain number of times rather than MULTIPLY. An expression with a negative exponent is the RECIPROCAL of what the expression would be with a positive exponent. Ex. (3/4)^-3 = (4/3)^3 = 64/27
What is a prime number? Is one a prime number?
A prime number is any number that is only divisible by 1 and itself that is a whole number greater than 1, so 1 is not a prime number. When listing the factors of a number to find its prime factors, do not count 1 as a prime factor.
Why is absolute value always a positive value?
Absolute value disregards the direction (left or right) from which the number approaches 0, and therefore disregards whether the number is positive or negative.
How do you find out if a larger number is a multiple of some other number?
Add the digits of the number together and see if the result is divisible by the other number. Ex. 252 is a multiple of 3 because 2+5+2=9 and 9 is divisble by 3.
If you multiply only odd integers together, what will be the result?
An ODD integer.
When you take the prime factorization of a number inside a square root, how can you determine which prime factors you can effectively bring out of the square root?
Any prime factor that can be paired off can be effectively brought out of the square root. Ex. the prime factorization of 360= 2×2×2×3×3×5, therefore the square root of 360= the square root of 2×2×2×3×3×5. 2 and 3 can be paired up, leaving 2×5 inside the square root. Bring 2 and 6 out of the square root and multiply to get 6. Multiply 2×5 inside the square root to get 10. Therefore, the square root of 360= 6 × the square root of 10.
As you raise a positive proper fraction to higher and higher powers, what happens to the result?
As the exponent increases, the value of the expression decreases. Ex. 3/4^2= 9/16
How can you use a factor tree to find the prime factors of a number?
Begin with the number you are factoring and then draw a line/branch from the number connecting each factor pair and then repeat the process for each number in each factor pair until you reach a point at which you can only see prime numbers. Circle the prime numbers to remind yourself that the branch cant break down any further. Don't continue to break down prime numbers into their only factor pair, one and the number itself.
When you see two terms with different bases being multiplied together or one being divided by the other, what should you check before trying to simplify?
Check to see if the bases share a common factor that can be multiplied out to produce a common base. Ex. 5^3 × 25^2 can be rewritten as 5^3 × 5^2)^2= 5^3 × 5^4= 5^7
As you raise decimals between 0 and 1 to higher and higher powers, what happens to the result?
Decimals between 0 and 1 decrease as their exponents increase. Ex. 0.6^2= 0.36
When asked to find the arithmetic mean of a set of numbers, what does that mean? How is that different than just calculating the average?
Finding the arithmetic mean of a an evenly spaced set of numbers with an odd number of terms means finding the middle value of the set. Ex. the arithmetic mean of 26, 28, 30= 28.
What question does the absolute value of a number answer?
How far away from 0 is the number on a number line? Ex. 5 and -5 have the same absolute value (5) because they are both 5 units away from 0 on a number line.
What does the factor foundation rule state?
If a is divisble by b, and b is divisble by c, then a will be divisble by both b and c. Ex. If you know that 12 is divisble by 6, and 6 is divisble by 3, then 12 is divisble by 3 as well. This rule also works in reverse to some extent: if d is divisble by two different primes, e and f, d is also divisble by e×f. Ex. If 20 is divisble by 2 and 5, it is also divisble by 2×5 or 10. (e and f can also represent the same prime factor as long as there are at least two copies of that prime factor in the prime factorization of d)
What is an example of the GRE testing the divisibility rules in reverse?
If you are told that a number's ones digit is equal to 0, then you the number is divisble by 10. Also, if you are told that the sum of the digits of a number (x) equal 21, you know that x is divisble by 3 but NOT by 9.
What is meant by "unique prime factors", also referred to as "distinct factors"?
If you prime factor 12, you get 2×2×3. While 12 has three prime factors, it only has two unique prime factors: 2 and 3. Another example would be 100 and 10. While the prime factors of 100 are 2×2×5×5 and the prime factors of 10 are 2×5, both have the same unique prime factors: 2 and 5.
What is meant by the mnemonic "fewer factors, more multiples"?
It helps you differentiate between factors and multiples. Factors divide into an integer and are therefore less than or equal to that integer. Positive multiples, on the other hand, multiply out from an integer and are therefore greater than or equal to that integer. An integer only has a limited number of factors, but there are an infinite amount of multiples of an integer. Factors can also be called divisors of an integer.
When an unknown variable has a positive exponent, what does the exponent essentially do?
It hides whether the variable's value is positive or negative since a negative and a positive number raised to a positive power will always be positive.
How can you tell if a number is divisble by 6?
It is divisble by both 2 and 3. Ex. 48 is divisble by 2 because it is an even number and by 3 because the sum of its digits is divisble by 3.
What is the sum of all of the elements in a sequence equal to?
It is equal to the arithmetic mean (average) × the number of items in the sequence.
Is one a prime number?
NO.
If x^2= 16, does x= 4?
NO. -4 is also a possibility. When a variable is being raised to an even exponent, it hides the sign of the base, so it could be negative.
If xy does not equal 0, is -x × -y definitely positive?
NO. Just because a variable has a negative sign in front of it does not mean it has a negative value. If x or y really represents a negative number, then the negative sign in front of the variable creates a double negative and the value of the variable would become positive. If both x and y are positive or negative numbers, xy would definitely be positive. But if one of them is really a negative number, xy would then be negative.
Can you further simplify 7^4 + 7^6?
NO. You cannot add exponents of expressions with the same base when they are being added, only when they are being multiplied.
Can you further simplify 12^7 - 3^7?
NO. You cannot rewrite the bases of expressions linked by addition/subtraction to pull out common bases, only expressions linked by multiplication/division.
If you see a sum of two primes that is odd, what can you determine about one of them?
One of the primes must be 2.
What are the most common prime numbers on the GRE?
Prime numbers less than 20: 2, 3, 5, 7, 11, 13, 17 and 19.
When multiplying and dividing two numbers, what rule (SSSNN) should you follow?
Same signs rule: If Signs are the Same, the answer is poSitive. If Not, then the answer is Negative. pos × pos= pos neg × neg= pos pos × neg= neg
How do you change a number being raised to a negative power to a positive power?
Switch the number to its inverse. Ex. (1/4)^-3 = 1/(1/4)^3
What is true about the average of any sequence of consecutive integers with an EVEN number of items?
The average will not be an integer. The average is always equal to the average of if the two middle terms, which is exactly between two consecutive integers. Therefore, the average will not be a whole number. Ex. (4+5+6+7+8+9)/6= 39/6= 6.5
What should you remember when you see a base without an exponent being multiplied/divided by the same base with an exponent?
The base without an exponent is really the base being raised to the power of 1, so if you see 3 × 3^4 it could be simplified to 3^4+1= 3^5. When you see a base without an exponent, you should write in the exponent 1.
How do you find the distance between two points on a number line if you know their specific locations?
The distance between the two points is the absolute value of their difference. Ex. point A is located at -3 on a number line and point B is located at 8. The distance between point A and point B= I8-(-3)I = 11.
Why are there no guarantees about the outcome of the result of division by even and odd integers?
The division of the two integers might not yield an integer result (the result may not be a whole number).
When multiplying exponential terms that share a common base, what happens to the exponents?
The exponents get added together. x^5 × x^3 = x^8
How is finding the number of MULTIPLES in a sequence of consecutive MULTIPLES different than the "add one before you're done" method of finding the number of consecutive INTEGERS in a sequence of INTEGERS?
The formula for finding the number of consecutive INTEGERS in a sequence is (last number-first number +1), but the formula for finding the number of consecutive MULTIPLES in a sequence is (last number - first number ÷ increment + 1).
How do you know if a number is divisble by 5?
The number has a 5 or 0 at the end.
How can you tell if a number is divisble by 8?
The number is divisble by 2 three times in succession, or if the three-digit number at the end is divisble by 8. Ex. 23,456 is divisble by 8 because 456 is divisble by 8.
How can you tell if a number is divisble by 4?
The number is divisble by 2 twice, or if the two-digit number at the end is divisble by 4.
How can you tell if a number is divisble by 3?
The number is divisble by 3 if the SUM OF ITS DIGITS is divisble by 3. Ex. 72 is divisble by 3 because the sum of its digits is 9, and 9 is divisble by 3.
When a question refers to points on a number line, what is always true about the position of points on the line?
The numbers on a number line get bigger as they go from left to right, therefore points will increase in value from left to right.
X is a non-negative even integer. Quantity A= X. Quantity B= 1. Which quantity is greater?
The relationship between QA and QB cannot determined. QA could be 0, because 0 is the first non-negative even integer, or it could be 2, 4, 6, 8... etc. Therefore, QA could be either greater than or less than QB, which is 1.
What is true about the range of possible remainders when dividing by a number?
The remainder must be an integer (while number) greater than 0 but less than the number you are dividing by.
What is the result of multiplying three negative numbers together?
The result is a negative number.
What is the result of adding or subtracting two even integers or two odd integers?
The result is an EVEN integer. 7+11=18 14-6=12
What is the result of adding or subtracting an odd and an even integer?
The result is an ODD integer. 7+8=15 12-5=13
What is true about raising 0 to the power of 0?
The result is indeterminate and this expression will never occur on the GRE. If you see a variable being raised to 0, the variable cannot equal 0.
What is true about the result of raising any base to the power of 0?
The result will always be 1. Ex. 3^0= 1.
What is the result of multiplying a positive and a negative number together?
The result will be a negative number.
What is the result of multiplying one positive number and two negative numbers together?
The result will be a positive number.
What is the result of multiplying two negative numbers together?
The result will be a positive number.
What is the result of multiplying two positive numbers together?
The result will be a positive number.
What is the result of multiplying ANY set of integers together that include one even integer?
The result will be an EVEN integer.
What is the result of adding or subtracting multiples of an integer (N)?
The result will be another multiple of N. Ex. if N is the divisor of x and y, N is the divisor of x+y.
What can you determine about the result of multiplying together a set of integers that includes two even integers?
The result will be divisble by 4 because the two even integers each contribute a 2 as a factor of the product. Ex. 2×5×6= 60
What can you determine about the result of multiplying together a set of integers including three even integers?
The result will be divisble by 8 because each even integer contributes a 2 to the factors of the product.
What can you determine about the result of multiplying several even integers together?
The result will be divisible by higher and higher powers of two.
What is true about the square root of a squared variable? What is the square root of x squared?
The square root of a squared variable is equal to the ABSOLUTE VALUE of of that variable. The square root of x squared is IxI, NOT x.
What can you determine about the sum of any two prime numbers not including 2?
The sum of any two primes will be EVEN unless one of them is 2. This is because all prime numbers except 2 are ODD.
How can you tell if a number is divisble by 9?
The sum of its digits is divisble by 9. Ex. 4,185 is divisble by 9 because 4+1+8+5= 18, and 18 is divisble by 9.
If you know that 2 cannot be one of two primes summed together, what can you determine about the sum?
The sum of the two primes must be even.
When something with an exponent is being raised to another power, what happens to the exponents?
They get multiplied together. (a^2)^4= a^8
When dividing exponential terms that share a common base, what happens to the exponents?
They get subtracted from one another. x^6/x^2= x^4
In order for a number or unknown variable to be divisble by another number, what must the number/variable have in common with the other number?
They must have all if the same prime factors.
What is the arithmetic mean of 4, 8, 12, 16, 20 and 24?
This is also an evenly spaced sequence of integers, but since there is an even number the middle number/arithmetic mean can be found by finding the average of the two middle numbers (adding them together and dividing by 2). It can also be found by finding the average of the smallest and largest numbers in the sequence. middle numbers: 12+16= 28/2= 14 smallest/largest: 4+24= 28/2= 14
What is the arithmetic mean of 4, 8, 12,16, and 20?
This is an evenly spaced sequence of integers, so the arithmetic mean is equal to the median. The middle number is 12, so 12 is the arithmetic mean. The arithmetic mean can also be found by finding the average of the smallest and largest numbers in the sequence (adding them together and dividing by 2). 4+20= 24/2= 12
What is the difference between raising a compound base to higher powers when the compound base consists of a product or sum/difference?
When the base of an exponential expression is a product, you can multiply the base together and then raise the result to the exponent OR you can raise each number in the expression to the exponent and then multiply. But when the compound base consists fo a sum/difference, you MUST perform the addition/subtraction in the base BEFORE raising the result to the exponent.
If you are told that x^6= x^8= x^10, what could x equal?
X could equal 1, 0 or -1 because the even exponents guide the sign of the base.
If you are told that x^6 = x^7 = x^15, what must x equal?
X must equal 1 or 0. The number 0 raised to any positive power yields 0, and 1 raised to any power yields 1. The answer cannot be -1 because -1 raised to an even power yields 1 and -1 when raised to an odd power. If you plug -1 into the equation, it will not hold true.
If there is more than one point on a number line, what can you determine about the points and what can you not determine?
You CAN determine RELATIVE POSITION of points on the line, but you CANNOT determine RELATIVE DISTANCE between the points unless you are specifically given that information. Ex. On a number line, points A, B, and C lie on the line from left to right. You can determine that A<B<C, but you can't determine how far apart any of the points are on the line. The rules are similar if a number line contains both numbers and variables.
How can you go about solving 7^4 + 7^6?
You can factor out a common base and exponent. Both expressions have 7^4 in common, so you can factor out to produce 7^4(7^2 + 1)= 7^4(50).
If you only know the prime factors of a number, what can you do to find larger factors of the number?
You can multiply the prime factors together in different combinations to find larger factors or the number. This is because all of the factors of a number (except for 1) can be built from combinations of its prime factors. Ex. the prime factors of 150 are 2×3×5×5, and you can see that is also divisble by 6, 10, 25 and 15.
When asked to multiply two roots together or divide one by the other, what can you do to solve the problem?
You can multiply the two numbers inside the roots together and then take the square root of the result. Ex. the square root of 8 × the square root of 2 can be rewritten as the square root of 8×2= the square root of 16= 4. You can also divide the number inside the root in the numerator by the number inside the root of the denominator and then take the square root of the quotient. Ex. the square root of 27 ÷ the square root of 3= the square root of 9= 3. These rules can also be used in combination within the same problem if it involves both multiplication and division.
When adding or subtracting exponents with the same base, what should you NOT do and what should you do instead?
You should NOT add simply add the original exponents together. When adding or subtracting exponents with the same base, it's not possible to directly combine exponents. FACTORING OUT is the correct procedure.
What is the best way to use a factor true to find the prime factorization of larger numbers (Ex. 630)?
You should begin with the smallest prime factors of the number and work your way up to larger ones. Ex. when making a factor tree for 630, start with the smallest prime factor that will go into 630, which would be 2. The first factor pair in your tree will be 2 and 315. To start factoring 315, see if it divisble by 2, 3, 5, etc down the smallest price factors list. It is not divisble by 3 because 3+1+5=9, so your next factor pair on the tree branching from 315 will be 3 and 105. Remember to circle prime factors as you go.
If given two variables and a factor of each variable and then asked if another number could be a factor of those two variables combined, what should you do to determine if this number is a factor of the product of the variables?
You should draw a factor tree for each variable and its factor and determine the prime factors, and then draw a factor tree for the possible factor of the combined variables to determine its prime factors. However, you should think of the prime factors of each variable as part of the same factor tree, so if the possible factor has prime factors in common with both of the variable's factor trees, it is a factor of their product.
What is a × a^x simplified?
a × a^x= a^x+1
What is a^x × a^x simplified?
a^x × a^x= 2a^x
What is the definition of evenly spaced sequences?
sequences of numbers whose values go up or down by the same amount (the increment) from one item to the next Ex. 4, 7, 10, 13, 16 is evenly spaced because each item goes up by 3 over the previous value
What is the definition of consecutive multiples?
special cases of evenly spaced sequences: all values in the sequence are multiples of the increment Ex. 12, 16, 20, 24 is a sequence of consecutive multiples because the values increase from one to the next by 4, and each element is a multiple of 4 *sequences of consecutive multiples MUST be composed of integers
What is the remainder formula in algebraic form?
x/N= Q+ R/N OR x= Q×N+R (Ex. 23= 5×4+3) x= dividend (numerator) N= divisor (denominator) Q= quotient (resulting integer portion that can be divided out) R= remainder
