GMAT Math 2

What is the Equation for remainder theory?

(x/y) = Q + (r/y) or x = Qy +r

Formula for Division

(x/y) = Q + R/y R= remainder

Divisibility with exponents

(x^a)/(x^b) = an even integer if a > b

Perfect Squares to Memorize

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.

Perfect Cubes to Memorize

0, 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1,000

What is the only number with exactly one factor?

1 (Properties of One)

Is one a prime number?

1 (Properties of One). No.The first prime number is 2.

Example 1: Is mn < 0 ? 1) m^5 * n^2 < 0 2) m^11 * p^8 * n^5 < 0

1) Not sufficient. As n is squared, we cannot get definitive information on the value of n; therefore, we cannot get information on nm < 0. 2) m and n are both raised to off powers. Thus, they will retain the sign of the original integer. This is enough information to decide mn < 0. Answer B.

Tips about Perfect Cubes

1) The cube root of a perfect cube will always be an integer. 2 )A perfect cubes, other than 0 or 1, have prime factors with exponents that are divisible by 3.

Tips about Perfect Squares

1) The square root of a perfect square will always be an integer. 2) all perfect squares must end in 0, 1, 4, 5, 6, or 9. 3) All perfect squares have prime factors with even exponents (except 0 and 1).

Division phrases that mean dividing (x/y; dividing x by y) will result in an integer

1) y is a factor of x 2) y is a divisor of x 3) y divides into x (evenly) 4) x is a multiple of y 5) x is a dividend of y 6) x is divisible by y

When a nonzero base is raised to the zero power, the expression equals...

1. Ex. 9^0 =1 BUT Ex. 0^0 = 1

Properties of One

1. One is the only number with exactly one factor. 2. One is not a prime number. (The first prime number is 2). 2. One is the smallest factor in every number (except 0).

Properties of Zero

1. Zero is neither positive nor negative. 2. Zero is even 3. Zero is the only number equal to its opposite (Ex. 0=0). 4. While Zero is a multiple of every number, zero is NOT a FACTOR of any number except zero (Ex. 7/0 is not possible) 5. 1 and -1 are the only rational numbers which are their own reciprocals.

Prime numbers less than 100

25 total: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Note: When an expression contains both radicals and non radicals, multiply/divide radicals by radicals (with the same index) and multiply/divide non-radicals by non-radicals.

2√10 * 4√7 = 8√70

Example P and Q are integers, and r is the remainder when P is divided by Q. If P = 3.16Q, which of the following must divide evenly into r? I. 2 II. 4 III. 8 1) I only 2) II only 3) I & II only 4) II & III only 5) I, II, III

3) I & II only Note: The smallest R can be is 4. R needs to divide evenly into 4. R can be 2 or 4/ I and II.

How do you get a Terminating Decimal?

A fraction with a denominator that has prime factors of ONLY 2s and 5s will have a decimal that terminates. Note: A denominator with any other prime factors produces decimals that do not terminate.

Multiple

A multiple of a number is the product of that number and any integer. Example: What are the multiples of 4? → 4,8,12,16,20,....,4n

Number Divisible by 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.

Number Divisible by 6

A number is divisible by 6 if the number is divisible by both 2 and 3.

Number Divisible by 9

A number is divisible by 9 if the sum of all the digits is divisible by 9.

Number Divisible by 12

A number is divisible of 12 if it is divisible by 3 and 4.

Example: At a certain department store, a rug was originally priced at W dollars, where W is a whole number. During a liquidation sale, the rug was sold for 6 percent of its original price. Which of the following could be the sale price of the rug? A) $27 B) $28 C) $29 D) $31 E) $32

A) $27

When 2/11 is expressed as a decimal, what is the 79th digit to the right of the decimal place? A) 1 B) 2 C) 3 D) 6 E) 8

A) 1 To solve: 2/11 = 1/11(2) 1/11 = 0.09090909 2/11= 0.090909(2) = 0.181818181818 80th = 8 79th = 1

20^2 + 21^2 + 22^2 + 23^2 + 24^2 + 25^2 = A) 3,055 B) 3,060 C) 3,066 D) 3,704 E) 3,077

A) 3,055 Shortcut: Focus on the unit digits = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 0 + 1 + 4 + 9 + 16 + 25 = 55

When taking the Square Root of a number and/or Binomial, the result is an...

Absolute value. Ex. √x^2 = |x| thus √(x+y)^2 = |(x+y)| Ex. (x+6)^2 = 49 √(x+6)^2 = √49 x+6 = |7| x = [1,-13]

How do you add and subtract remainders?

Adding and Subtracting Remainders Remainders can be added and subtracted, but we need to correct the excess at the end. Ex. (12 + 13 + 17)/5. 12/5 = 2 + 2/5 13/5 = 2 + 3/5 17/5 = 3 + 2/5 R = 2 + 3 + 2 = 7 R = 7/5 R = 7 - 5 R = 2

Example: If 5^2, 3^2, and 2^2 are factors of 150X, what is the smallest positive possible value of X? Note: 150 = 2, 5, 5, 3 A) 1 B) 6 C) 12 D) 18 E) 24

B) 6

What is the largest integer value of k, such that [400! / 5^k ] is an integer? A) 100 B) 99 C) 96 D) 85 E) 80

B) 99 To solve: 400/5= 80 400/25 = 16 400/125 = 3.... 400/625 = 0 80+16+3 = 99 The largest integer value of k = 99.

How many zeros are to the right of the last non-zero digit in the number 5^18 × 2^20? A) 16 B) 17 C) 18 D) 19 E) 20

C) 18 First thing I noticed, there are 18 (5 x 2) pairs. The answer must be 4 (or 2x2) with 18 trailing zeros.

If (90! / 15^n ) is an integer, what is the largest possible value of integer n? A) 18 B) 20 C) 21 D) 22 E) 24

C) 21 To solve: 90! / 15^n = 90! / 5^n * 3^n Note: 5 is the limiting factor 90/5 = 18 90/25 = 3... 90/125 = 0... =18 + 3 =21 pairs of (5x3)

What is the largest number that must be a factor of the product of any four consecutive positive integers? A) 6 B) 12 C) 24 D) 30 E) 48

C) 24 The product of any set of n consecutive, positive integers is divisible by n! Ex. 5 x 6 is divisible by 2! (30 is divisible by 2) Ex. 7 x 8 x 9 is divisible by 3! (504 is divisible by 6) Ex. 3 x 4 x 5 x 6 is divisible by 4! (360 is divisible by 24)

Example: The least common multiple of 8, 10, and x is 120. All of the following could be values of x except for which of the following. A) 3 B) 6 C) 9 D) 12 E) 15

C) 9 120 does not have 3^2 in its prime factorization.

What is the remainder when 3^123 is divided by 4? A) 0 B) 1 C) 2 D) 3 E) 4

D) 3 To solve: 3^1 / 4 = R3 3^2 / 4 = R1 3^3 / 4 = R3 3^4 / 4 = R1 3^123 fits into the category (3^3 / 4 = R3). So, it will also have a remainder of 3

How many leading zeros (zeros after the decimal point but before the first nonzero digit) are in the number 1/[5^5 × 2^11]? A) 3 B) 4 C) 5 D) 6 E) 8

D) 6 = 1/ [5 x 2]^5 * 2^6 =1/ [10]^5 * 64 k= 7 digits leading zeros = k - 1 leading zeros = 6

There is a certain number of students in Mr. Stewart's class. Could Mr. Stewart evenly divide the class into 3 study groups? 1) Mr. Stewart reduced the number of students in his class by 6 percent he could evenly divide the class into groups of 3. 2) Mr. Stewart reduced the number of students in his class by 16 percent he could evenly divide the class into groups of 9. A) 1 is enough, 2 is not B) 2 is enough, 1 is not C) Both together are enough D) Each alone is enough E) Neither is enough

D) Each alone is enough (Flash card Q)

Absolute value

Distance away from zero

Example If a, b, d, and x are positive integers, and x = 3abd, all of the following must be factors of x except: A) a B) ab C) ad D) 3d E) 2a

E) 2a Note: Divide x to both sides. In order for the quotient to be an integer, x must be a factor of any combination of 3abd

Addition and Subtraction Rules for Even and Odd Numbers

Even (+/-) Even = Even Odd (+/-) Odd = Even Even (+/-) Odd = Odd

Division Rules for Even and Odd Numbers

Even / Odd = Even 12/ 3 = 4 Odd / Odd = Odd Ex. 15 / 5 = 3 Even / Even = Odd or even Ex. 12/6 = 2 BUT 12/4 = 3 **Not possible for integers** Odd/Even

What is an even division?

Even Division: means dividing (x/y or dividing x by y) will result in an integer.

Even and Odd Numbers Formula

Even integers: 2n Odd integers: 2n + 1

Multiplication Rules for Even and Odd Numbers

Even x Even = Even Odd x Odd = Odd Even x Odd = Even *Note: The product of an even number and any integer is always even.

How do you determine the largest number of non-prime number x that divides into y! ? Ex. (40!/6^n

Ex. (40!/6^n) or (40!/2*3)^k Shortcut: Divide 40 by 3^k) ; 3 is the limiting factor 40/3 = 13... 40/9 = 4... 40/27 = 1... 40/81 = 0... 13+4+1 = 18 There are 18 threes in 40. The maximum number of (2 x 3) pairs in 40! is 18.

Watchout: The square root of a negative number is not a real number

Ex. √-100 Note: It is possible to take the off root of a negative number

How do you determine the largest number of a prime number x that divides into y! ? Ex: [400! / 5^n ]

Given Ex. [400! / 5^n] To determine the largest number of a prime number x that divides into y! : 1) Divide y by x1, x2, x3 until y/x^k produces a quotient of zero. 2) Add the quotients from the previous divisions; sum represents the total prime numbers

How do you determine the number of leading zeros in a fraction, 1/x

How to Determine the number of Leading Zeros 1) Express the fraction in the form 1/x. (x is an integer) 2) If 1 < x <=10, NO Leading zeros 3) If 10 < x <= 100, ONE Leading zero 4) If 100 < x <= 1000, TWO Leading zero If X is an integer with k digits, (1/X) will have k-1 leading zeros. If x is a perfect power of 10, or (10^n), there will be k-2 leading zeros.

Number Divisible by 8

If the last three digits of a number are a number divisible by 8, then the number is divisible by 8.

Number Divisible by 4

If the last two digits of a number are a number divisible by 4, then the number is divisible by 4. Ex. 28 Ex. 168 Ex. 00

Factor

If y divides evenly into x, we say y is a factor of x. Example: What are the factors of 16? → 1, 2, 4, 8,and 16

How can we identify trailing zeros?

In whole numbers, trailing zeros are created by (5 × 2) pairs.

What is the range of possible remainders?

Integers: (0, n-1) Ex. 72/3 n = 3 largest remainder: 2 or (n - 1 or 3 - 1) smallest remainder: 0

LCM(100, 25) GCF(100, 25)

LCM(100, 25) = 100 GCF(100, 25) = 25

What are leading zeros?

Leading zeros are zeros to the right of a decimal point. Ex. 2/10 = 0.2 -- No leading zeros 2/100 = 0.02 -- 1 leading zero 2/1000 = 0.002 -- 2 leading zeros

Remember: All you need is one even number in (6xyzabc) for the product to be even.

Multiplication Rules for Even and Odd Numbers

Any factorial > or = 5! will always have a zero as a units digit.

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Can you convert a decimal remainder to fraction form?

Not really When division of two integers (x and y) results in a decimal, Ex. (x/y) = 9.48 The actual remainder must be a multiple of the most reduced fractional remainder. Ex 48/100, 24/50, or 12/25. All we know if that the remainder is a multiple of 12.

Addition and Subtraction of Radicals

Only like radicals can be added to or subtracted from one another. √a + √b ≠ √a+b Example: √25+16 ≠ 5+4 BUT √25+16= √41 ≈ 6.40

Finding the Total Number of Factors in a Particular Number

REMEMBER: To find the total number of factors: 1) Find the prime factorization 2) Add one to each exponent 3) Multiple the exponents. The product = the total number of factors. **For the odd factors. Only add & multiply the odd prime numbers **For the even exponents: [all - odd].

Least Common Multiple (LCM)

The LCM of any set of positive integers is the smallest positive integer into which all of the numbers in the set will divide. (they have the same smallest prime factorizations) Ex LCM(2,5) = 10

Greatest Common Factor (GCF)

The largest factor that two or more numbers have in common. GCF of (8, 12, 16) is 4.

Number Divisible by 3

The sum of the digits is divisible by 3 Ex. 243

How do you convert a decimal remainder to an integer?

To Convert a decimal remainder to an integer, you can multiple the decimal component (0.8) by the divisor (5) to get 0.8 * 5 = 4. To employ this strategy, you need to know the denominator. Ex. (9/5) = 1.8 0.8 (5) = 4. Remainder = 4 OR 9/5 = 1 + 4/5

How to use trailing zeros to determine the number of digits in an integer?

To determine units digit in a number 1) Prime factorize 2) count the number of (5 x 2) pairs. Each pair contributes to one trailing zero. 3) Collect the unpaired 5s/2s/or other nonzero prime factors and multiply them. Count the number of digits in this product. 4) Sum the number of digits from steps 2 and 3.

How to Find the GCF?

To find GCF: 1) Find prime factorization of each integer. 2)Take the lowest of any repeated factor Ex. If we had (3^2) and (3^3), we would take (3^2). 3) Multiply all the integers found. The product = GCF Note: If there are no repeating integers, the GCF = 1)

How to find the LCM?

To find LCM: 1) Find prime factorization of each integer. 2)Take the highest of any repeated factor Ex. If we had (3^2) and (3^3), we would take (3^3. 3) Take all the non-repeated prime factors of the integers. 4) Multiply all the integers found. Or If two positive integers, x and y, share no prime factors, the LCM of x and y is xy.

Total number of prime factors Vs. Unique Prime Factors

Total number of prime factors = counting all the prime factors in the prime factorization tree. Unique prime factors: counting the number of prime factors that differ from each other. 6: 2x3; 2 unique prime factors; 2 total prime factors 12: 2x2x3; 2 unique prime factors; 3 total prime factors. 32: 2x2x2x2x2; 1 unique prime factors; 5 total prime factors

If x is a positive integer, what is the units digit of 3^(16x+18)? A) 9 B) 7 C) 3 D) 2 E) 1

Trick: Split the exponent 3^(16x+18) = [3^(16)]^x * 3^(18) Units digit of 3 3^1 =3 3^2=9 3^3=7 3^4= 1 3^(16) = units digit of 1 3^(16) = units digit of 9 1*9 = 9

How do you handle multiple square roots?

Trick: Treat each number, n, separately by raising n to a power of (1/2) for every square root it is under

True or False: If z is divisible by both x and y, z must also be divisible by the LCM of x and y.

True If (z / x ) = integer & (z / y ) = integer, then [z / LCM(x,y) ] = integer. If z is divisible by both x and y, z must also be divisible by the LCM of x and y.

True or False? if (x/y) is an integer, then (x/any factor of y) is an integer.

True. If x and y are positive integers and (x/y) is an integer, then (x/any factor of y) is also an integer.

Remember: Two consecutive integers will never share the same prime factors.

Two consecutive integers will never share the same prime factors. Thus, GCF of consecutive integers is 1. Or, GCF(n, n+1) = 1.

Remainder after Division: 10

When a whole number is divided by 10, the remainder will be the units digit of the dividend (numerator). Ex. 53/10 R= 3

Remainder After Division: 100

When a whole number is divided by 100, the remainder will be the last two digits of the dividend, etc. Ex. 153/100 R= 53

What are the remainders after Division by 5?

When integers with the same units digit are divided by 5, the remainder is constant. Ex. R=2 : 7/5, 17/5, 27/7, 37/5 R=4 : 9/5, 19/5, 29/5, 39/5, 49/5

Can the LCM of a set of positive integers tells us the unique prime factors of the numbers in the set?

Yes. LCM of a set of positive integers tells is all of the unique prime factors of the numbers in the set. We can also find the unique prime factors of the product of the numbers in the set.

Can you multiply remainders?

Yes. Remainders can be multiplied, but we need to correct the excess at the end. Ex. (12 * 13 * 17)/5. 12/5 = 2 + 2/5 13/5 = 2 + 3/5 17/5 = 3 + 2/5 R = 2 * 3 * 2 = 12 R = 12/5 R = 2 + 2/5 R = 2

If we know the LCM and the GCF of two positive integers, X and Y, do we know the Product of X and y?

Yes.That is, xy = LCM(x, y) × GCF(x, y).

What number is neither positive nor negative?

Zero. Properties of Zero

What number is the only number equal to its opposites?

Zero. Properties of Zero.

Multiplication of Like Bases means that exponents get...

added.

When dividing exponential #s with a different base and the same exponent, you do this to the exponents...

keep the common exponent and divide the bases.

When multiplying exponential #s with a different base and the same exponent, you do this to the exponents...

keep the common exponent and multiply the bases.

When exponents are raised to another exponent, the exponents get...

multiplied

Note: Order of operations stipulates that we perform any addition or subtraction under a radical prior to taking the root.

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REMEMBER: Prime numbers do have a prime factor, the prime number itself.

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Remember: All positive integers, regardless of their size, have a finite number of factors.

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WATCH OUT: On GMAT, radicals must be removed from the denominator in order for the expression to be considered simplified.

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Division of like bases means that exponents get...

subtracted.

When a base is raised to the first power, the value of the expression is simply...

that base. Ex. 5^1 = 5

Interpret: "a number x is a multiple of 5"

x = 5n Note: (x/5) is an integer

√2≈?, √3≈?, √5≈?

√2≈1.4, √3≈1.7, √5≈2.2

Dividing Radicals

√a/b = √a/√b Ex. √54/ √6 = √54/6 = √9 = 3 Beware: Be sure to never combine radicals with a different index

Multiplying Radicals

√ab = √a x √b or √5× √7= √35 Note: You can only multiply radicals with the same index