GRE 2022 (Math Foundations-Arithmetic And Number Properties Review)

6,930 is a multiple of what numbers?

2, 3, 5, 6, 9

(2)What is the value of |6-4x2|?

2 First multiply 4 by 2. The expression is now |6-8|. Then subtract 8 from 6 to get -2. Finally, take the absolute value of -2 to get 2.

(5)If a, b, and c are positive integers and a and c are odd, what is the smallest possible value of b given a x b x c is even?

2 When multiplying integers, at least one integer must be even to get an even result. If a and c are odd, then b must be even for a x b x c to be even. That means the smallest value of b, which must be positive, is 2.

Exponent and Roots

(5^0)=1 (161^0)=1 (-6^0)=1

66(3-2)/11=

66(3-2)/11= 66(1)/11= 66/11= 6

(0.029)10^3=

(0.029)10^3= 0029.=29

(1/2)^2=

(1/2)(1/2)=(1/4)

(12√15)/(2√5)=

(12√15)/(2√5)= (12/2)(√15/√5)= 6(√15/√5)= 6√3

(6√3)2√5=

(6√3)2√5= (6)(2)(√3)(√5)= 12√15

√75=

√75= (√36)√2= 6√2

4^3

(4)(4)(4)=64

(-2)^3=

-8

0.029x10^6=

0.029x10^6=0029000.=29,000

5/8=

0.625

(15)What is the value of (^3√.125)^(-4)?

16 Start by converting .125 to a fraction. Once that's done, use the rules of radicals and exponents to simplify: (^3√.125)^(-4)= (^3√(1/8))^(-4)= ((^3√2)/(^3√8))^(-4)= (1/2)^(-4)= 1/((1/2)^4)= 1/((1^4)/(2^4))= 1/(1/16)= 1x(16/1)= 16

2√3+4√2-√2-3√2=

2√3+4√2-√2-3√2= (√4-√2)+(2√3-3√3)= 3√2+(-√3)= 3√2-√3

3, 12, and 90 are all multiples of what number?

3 3=(3)(1) 12=(3)(4) 90=(3)(30)

How many digits does 542 have?

3 digits 5,4, and 2

30-5(4)+[((7-3)^2)/8]=

30-5(4)+[((7-3)^2)/8]= 30-5(4)+[(4^2)/8]= 30-5(4)+(16/8)= 30-20+2= 10+2= 12

315.246

3=hundreds 1=tens 5=units .=decimal point 2=tenths 4=hundredths 6=thousandths

(-2)^2=

4

(6)How many positive factors of 54 are odd?

4 The positive factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54. Of those, four (1, 3, 9, and 27) are odd.

(8)What is the value of ((4^3)x(9^3))/(6^5)?

6 Whenever there are exponents in a fraction, see if it is possible to get a common base in both the numerator and the denominator. Here, the denominator has a base of 6. There are two ways to simplify the numerator to get a base of 6: ((4^3)x(9^3))/(6^5)= ((4x9)^3)/(6^5)=(36^3)/(6^5)= ((6^2)^3)/(6^5)= (6^6)/(6^5) or ((4^3)x(9^3))/(6^5)= [((2^2)^3)x((3^2)^3)]/(6^5)= ((2^6)x(3^6))/(6^5)= ((2x3)^6)/(6^5)= (6^6)/(6^5) Either way, the final result is (6^6)/(6^5)=(6^6)^(-5)=(6^1)=6.

How many digits does 321,321,000 have?

9 digits, but only 4 distinct (different) digits: 3, 2, 1, and 0

Associative Laws of Addition and Multiplication

Addition and multiplication are also associative; regrouping the numbers does not affect the result. Example: (3+5)+8=3+(5+8) 8+8=3+13 16=16 (a+b)+c=a+(b+c) (ab)c=a(bc)

Operation

A function or process performed on one or more numbers. The four basic arithmetic operations are addition, subtraction, multiplication, and division.

Integer

A number without fractional or decimal part, including positive and negative whole numbers and zero. All integers are multiples of 1. The following are example of integers -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5.

Part

A specified number of the equal sections that compose a whole.

Set

A well defined collection of items, typically numbers, objects, or events. The bracket symbols { } are normally used to define sets of numbers. For example, {2,4,6,8} is a set of numbers.

Communicative Laws of Addition and Multiplication

Addition and multiplication are both communicative, which means that switching the order of any two numbers being added or multiplied together does not affect the result. Examples: 5+8=8+5 (2)(3)(6)=(6)(3)(2) a+b=b+a xyz=zyx Division and subtraction are not communicative; switching the order of the numbers changes the result. For instance, (3-2)doesn't=(2-3); the left side yields a difference of 1, while the right side yields a difference of -1. Similarly, (6/2)doesn't=(2/6); the left side equals 3, while the right side equals (1/3).

Sequence

An order list of terms. The terms of a sequence are often indicated by a letter with a subscript indicating the position of the number in the sequence. For instance, a3 denotes the third number in a sequence, while aN indicates the Nth term in a sequence.

Distinct

Different from each other. For example, 12 has three prime factors (2, 2, and 3) but only two distinct prime factors (2 and 3).

What is the least common multiple of 6 and 8?

Start by finding the prime factors of 6 and 8 6=(2)(3) 8=(2)(2)(2) The factor 2 appears three times in prime factorization of 8, while 3 appears as only a single factor of 6. So the least common multiple of 6 and 8 is (2)(2)(2)(3), or 24. (2)(2)(2)(3)= 24

-3, 0, 3, 6, 9

Is a series of consecutive multiple of 3.

2, 3, 5, 7, 11

Is a series of consecutive prime numbers.

-4, -2, 0, 2, 4, 6

Series of consecutive even numbers.

The Distributive Law

The distributive law of multiplication allows you to "distribute" a facto over numbers that are added or subtracted. You do this by multiplying that factor by each number in the group. Example: 4(3+7)=(4)(3)+(4)(7) 4(10)=12+28 40=40 a(b+c)=ab+ac The law works for the numerator in division as well. (a+b)/c=(a/c)+(b/c) However, when the sum or difference is in the denominator--that is, when you're dividing by a sum or difference--no distribution is possible. 9/(4+5) is not equal to (9/4)+(9/5).

Exponent

The number that denotes the power to which another number or variable is raised. The exponent is typically written as a superscript to a number. For example, 3^3 equals (5)(5)(5). The exponent is also occasionally referred to as a "power." For example, 5^3 can be described as "5 to the 3rd power." The product, 125, is "the 3rd power of 5." Exponents may be positive or negative integers or fractions, and they may include variables.

Square

The product of a number multiplied by itself. A squared number has been raised to the 2nd power. For example, 4^2=(4)(4)=16, so 16 is the square of 4.

Denominator

The quantity in the bottom of a friction.

Exponent and Roots

(2^2)x(2^3)=(2x2)(2x2x2)=(2^5) or (2^2)x(2^3)=(2^(2+3))=(2^5)

Exponents and Roots

(3^2)(5^2)=(3x3)(5x5)=(3x5)(3x5)=(3x5)^2=(15^2)

Exponents and Roots

(3^2)^4=(3x3)^4 or (3^2)^4=(3x3)(3x3)(3x3)(3x3) or (3^2)^4=(3^(2x4))=(3^8)

Exponent and Roots

(4^5)/(4^2)=[(4x4x4x4x4)/(4x4)]=(4^3) or (4^5)/(4^2)=(4^(5-2))=(4^3)

(9)When integer a is divided by 5, the remainder is 2. When integer b is divided by 5, the remainder is 3. What is the remainder when a x b is divided by 5?

1 Numbers that leave a remainder of 2 when divided by 5 numbers that are 2 greater than a multiple of 5, e.g., 7(5+2), 12(10+2), 17(15+2)... Similarly, numbers that leave a remainder of 3 are 3 greater than a multiple of 5, e.g., 8, 13, 18... Pick any two valid numbers to test a x b. If a=7 and b=8, then axb=56. When 56 is divided by 5, the result is 11 with remainder of 1 (as 56 is 1 greater than 55, a multiple of 5). This will work out for any values of a and b. On Test Day, it may be tempting to try a second set of values. However, when the correct answer is a single number, there is no need to do so. The GRE will not play tricks, and every valid set of numbers will lead to the same result. Have confidence with the numbers you selected and move on.

5.6x10^6

(5.6x10^6)=5,600,000, or 5.6 million

Exponent and Roots

(5^-3)=(1/(5^3))=(1/125)

(8^5)= (5^8)=

(8^5)=32,768 an even number (5^8)=390,625 an odd number

(12)What is the value of (√3+3√27)(2√3-√27)?

-30 (√3+3√27)(2√3-√27)= √3(2√3)+√3(-√27)+3√27(2√3)+3√27(-√27)= 2(3)-√81+6√81-3(27)=6-9+6(9)-81= -3+54+81=-30

(1)What is the value of 6(-3+1)-6(3-4)?

-6 6(-3+1)-6(3-4)=6(-2)-6(-1)=-12-(-6)=-12+6=-6

What are the positive factors of 36?

1, 2, 3, 4, 6, 9, 12, 18, and 36 Can group these factors in pairs: (1)(36)=(2)(18)=(3)(12)=(4)(9)=(6)(6)

(14)If the digits of integer x are reserved and the resulting number is added to the original x, the sum is 7,777. What is the smallest possible value of x?

1,076 When the digits are revealed, the new number will have the same number of digits. No two 3-digit numbers add up to 7,777 (as the largest 3-digit number is 999, and 999 plus 999 is only 1,998), so x must be a 4-digit number. Let A, B, C, and D represent the digits of x. Adding x to its reverse would look like this. ABCD+DCBA=7777 The smallest 4-digit numbers would begin with 1, so A should be 1. 1BCD+DCB1=7777 For the sum to end in 7, D would have to be 6. 1BC6+6CB1=7777 After that, the smallest possible value of B would be 0. In that case, C would have to be 7. 1076+6701=7777 Thus, 1,076 is the smallest possible value of x.

2.4, 12, and 132 are all multiples of what number?

1.2 2.4=(1.2)(2) 12=(1.2)(10) 132=(1.2)(110)

(7)What is the largest prime factor of 46,000?

23 Any multiple of 10 will have prime factors of 5 x 2. In this case, 46,000=46x10x10x10=23x2x5x2x5x2x5x2. The number 23 is prime and cannot be broken down further.

(4)How many two-digit multiples of 6 are multiples of 15?

3 The least common multiple of 6 and 15 is 30. Every multiple of 30 will also be a multiple of both 6 and 15. That means there are only three two-digit ,multiples of both 6 and 15: 30, 60, 90.

What is the greatest common factor(GCF) of 36 and 48?

36=(2)(2)(3)(3) 48=(2)(2)(2)(2)(3) They have in common two 2's and one 3, so the GCF is (2)(2)(3)=12 Answer: 12

(3)What is the value of (2/3)/(1/6)?

4 Dividing by a friction is equal to multiplying by the reciprocal of that friction. So, dividing by (1/6) is equal to multiplying by 6/1. (2/3)x(6/1)=(12/3)=4

(10)If x is an integer, how many values of x are there such that |x|<6 and |x|>3?

4 |x| is the distance between 0 and x on a number line. If |x|<6, then x must be less than 6 units away from 0. If x is an integer, that means it is any integer from -5 to 5. If |x|>3, then x must be 4 or greater, or it can be -4 or less. To satisfy both conditions, x can be -5, -4, 4, or 5.

416.03/10,000=

416.03/10,000=.041603=0.041603

(13)What is the value of √63x(^3√56)x(7^(1/6))?

42 The mixture of radicals and exponents can be confusing. Exponents are usually easier to deal with, so start by converting each radical into an exponential term: √63x(^3√56)x(7^(1/6))=(63^(1/2))x(56^(1/3))x(7^(1/6)) 63, 56, and 7 are all multiples of 7, so factor out 7 from each term: (63^(1/2))x(56^(1/3))x(7^(1/6))=(9^(1/2))(7^(1/2))x(8^(1/3))(7^(1/3))x(7^(1/6)) (9^(1/2))=√9=3 and (8^(1/3))=(^3√8)=2. Use the rules of exponents to finish: 3(7^(1/2))x2(7^(1/3))x(7^(1/6))=(3x2)x[(7^(1/2))x(7^(1/3))x(7^(1/6))]= 6x7^[(1/2)+(1/3)+(1/6)]=6+7^[(3/6)+(2/6)+(1/6)]=6x7^1=42

(11)What is the largest 4-digit multiple of 71?

9,940 The largest 4-digit number is 9,999. When 9,999 is divided by 71, the result is 140 with a remainder of 59. That means 9,999 is 59 greater than the nearest multiple of 71. Subtracting 59 from 9,999 will produce the greatest 4-digit multiple of 71. Another way to approach this, after doing the division, is to recognize that 71x141 would produce a number larger than 9,999, so the largest 4-digit multiple of 71 would be 71x140.

Whole

A quantity that is regarded as a complete unit.

Number line

A straight line , extending infinity in either direction, on which numbers are represented as points. The number line below shows the integers from -3 to4. Decimals and fractions can also be depicted on a number line, as can irrational numbers, such as √(2). -2.5. 1/3 √(2) π <---------------------------------------------> -3. -2. -1. 0. 1. 2. 3. 4 The values of numbers get larger as you move to the right along the number line. Number to the right of zero are positive; numbers to the left of zero are negative. Zero is neither positive nor negative. Any positive number is larger than any negative number. For example, -300<4.

Properties of Zero

Adding zero to or subtracting zero from a number does not change the number. x+0=x 0+x=x x-0=x Examples: 5+0=5 0+(-3)=-3 4-0=4 Examples: Product of zero and any number is zero. (0)(z)=0 (z)(0)=0 (0)(12)=0

Prime Factors

Example: What is the prime factorization of 210? 210=(2)(105) Since 105 is odd, it can't contain another factor of 2. The next smallest prime number is 3. The digits of 105 added up to 6, which is a multiple of 3, so 3 is a factor of 105. 210=(2)(3)(35) The digits of 35 add up to 8, which is not a multiple of 3. But 35 ends in 5, so it is a multiple of the next largest prime number, 5. 210=(2)(3)(5)(7) Since 7 is a prime number, this equation expresses the complete prime factorization of 210. Example: What is the prime factorization of 1,050? (1,050) (105) (10) (21) (5) (5) (2) (7) (3) The distinct prime factors of 1,050 are therefore 2, 5, 3, and 7, with the prime number 5 occurring twice in the prime factorization. We usually write out the prime factorization by putting the prime numbers in increasing order. Here, that would be (2)(3)(5)(5)(7). The prime factorization can also be expressed in exponential form: (2)(3)(5^2)(7).

Properties of 1 and -1

Multiplying or dividing a number by 1 does not change the number. (a)(1)=a (1)(a)=a a/1=a Examples: (4)(1)=4 (1)(-5)=-5 (-7)/1=-7 Multiplying or dividing a nonzero number by -1 changes the sign of the number. (a)(-1)=-a (-1)(a)=-a a/-1=-a Examples: (6)(-1)=-6 (-3)(-1)=3 (-8)/(-1)=8

Absolut Value

The absolute value of number is the value of number without its sign. It is written as two vertical lines, one on either side of the number and its sign. Example: |-3|=|+3|=3 The absolute value of a number can be thought of as the number's distance from zero on the number line. Since both 3 and -3 are 3 units from 0, each has an absolute value of 3. If you are told that |x|=5, x could equal 5 or -5.

Friction

The division of a part by a whole. Part/Whole=Friction. For example, 3/5 is a fraction.

Numerator

The quantity in the top of a fraction.

Sum

The result of addition.

Product

The result of multiplication.

Difference

The result of subtraction.