Arithmetic
The Foundation of Your Math Skills
Arithmetic is the foundation of GRE math—it must be solid to support everything else. This section covers approximately 25% of Quantitative Reasoning questions.
- Properties of integers, prime numbers, and divisibility
- Fractions, decimals, and percentages
- Ratios, rates, and proportions
- Exponents, roots, and absolute value
- Number sequences and series
Integers and Prime Factorization
What Are Integers?
Integers include positive and negative whole numbers and zero: ..., -3, -2, -1, 0, 1, 2, 3, ...
A prime number is an integer greater than 1 with only two divisors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...
Prime Factorization
Every integer greater than 1 can be uniquely expressed as a prime factorization.
Example:
Find the prime factorization of 800:
800 = 2 × 400
= 2 × 2 × 200
= 2 × 2 × 2 × 100
= 2 × 2 × 2 × 2 × 50
= 2 × 2 × 2 × 2 × 2 × 25
= 2 × 2 × 2 × 2 × 2 × 5 × 5
800 = 25 × 52
Key Applications:
- Finding greatest common factor (GCF) and least common multiple (LCM)
- Simplifying radicals and fractions
- Determining number of factors of an integer
Division and Remainders
The Division Algorithm
When an integer c is divided by a positive integer d, the result is a quotient q and a remainder r, where:
c = qd + r
Or equivalently:
r = c - qd
The remainder r must satisfy: 0 ≤ r < d
Example:
100 divided by 45 is:
- Quotient q = 2 (because 2 × 45 = 90)
- Remainder r = 10 (because 100 - 90 = 10)
- Answer: 2 remainder 10
We can verify: 100 = 2(45) + 10 ✓
Percent Change
Percent Increase Formula
To find a percent increase, divide the amount of increase by the original base amount:
Percent Increase = (New - Original) / Original × 100%
Example:
If a quantity increases from 600 to 750, what is the percent increase?
- Increase = 750 - 600 = 150
- Percent increase = 150 ÷ 600 = 0.25
- Answer: 25%
Percent Decrease Formula
Percent Decrease = (Original - New) / Original × 100%
Fractions, Decimals, and Percentages
Fractions
1/2
Part/Whole
Decimals
0.5
Base 10 form
Percentages
50%
Per hundred
Common Conversions to Memorize
1/2 = 0.5 = 50%
1/3 ≈ 0.333... = 33.33%
1/4 = 0.25 = 25%
1/5 = 0.2 = 20%
1/8 = 0.125 = 12.5%
2/3 ≈ 0.666... = 66.67%
3/4 = 0.75 = 75%
1/10 = 0.1 = 10%
Ratios and Proportions
Understanding Ratios
A ratio compares two quantities. If the ratio of A to B is 3:2, this means:
- For every 3 units of A, there are 2 units of B
- A = 3x and B = 2x for some value x (the multiplier)
- The total is 5x
Example:
The ratio of boys to girls in a class is 3:5. If there are 24 students total, how many are boys?
- Boys = 3x, Girls = 5x
- Total = 3x + 5x = 8x = 24
- Therefore x = 3
- Boys = 3x = 3(3) = 9
- Answer: 9 boys
Exponents and Roots
Key Exponent Rules
xa · xb = xa+b
xa / xb = xa-b
(xa)b = xab
x0 = 1 (x ≠ 0)
x-a = 1/xa
(xy)a = xaya
Square Roots and Radicals
- √(ab) = √a · √b
- √(a/b) = √a / √b
- √(x2) = |x| (absolute value)
- Perfect squares to know: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Ready to Practice Arithmetic?
Master these foundational concepts with targeted practice questions.