﻿ GMAT Study Guide - Number Theory Review and Basic Formulas

# GMAT Study Guide - Number Theory Review and Basic Formulas

### Prime Numbers to 100

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

Only positive numbers can be prime

### Odds and Evens

even ± even = even

even ± odd = odd

odd ± odd = even

even * even = even

even * odd = even

odd * odd = odd

Zero is even.

### Positive and Negative Numbers

positive * positive = positive

positive * negative = negative

negative * negative = positive

positive / positive = positive

positive / negative = negative

negative / negative = positive

Zero is neither positive nor negative, however when GMAT refers to a "non-negative" number that number might be zero.

### Divisibility Shortcuts

2: The last digit is even.

3: The sum of the digits is divisible by 3.

4: The last two digits form a number divisible by 4.

5: The last digit is 0 or 5.

6: The number is divisible by both 2 and 3.

9: The sum of the digits is divisible by 9.

10: The last digit is 0.

12: The number is divisible by both 3 and 4.

### Fractions

irrational number: cannot be expressed as a fraction

numerator: The number on top of a fraction.

denominator: The number on the bottom of a fraction.

To add/subtract fractions with the same denominator: Add/subtract the numerator and place over the common denominator.

To add/sub fractions with different denominators: Find the least common denominator and convert the fractions.

To multiply fractions: Multiply the numerators and denominators.

To divide fractions: Invert the divisor fraction (find its reciprocal) then multiply.

### Exponents

\$x^0 = 1\$

\$x^1 = x\$

\$0^n = 0\$

\$0^{-n} = undefined\$

\$1^x = 1\$

\$x^{-n} = 1/x^n\$

\$x^n * y^n = (xy)^n\$

\$(xy)^n = x^n * y^n\$

\$x^n/y^n = (x/y)^n\$

\$(x/y)^n = x^n/y^n\$

\$x^n/y^n = (x/y)^n\$

\$(x^m)^n = x^{mn}\$

\$x^n * x^m = x^{n+m}\$

\$x^{n+m} = x^n * x^m\$

\$x^n/x^m = x^{n-m}\$

\$x^{n-m} = x^n/x^m\$

\$x^{1/n} = √^nx\$

\$x^{m/n} = √^nx^m\$

\$x^2 = 25\$; solve for x; \$x = ±5\$

\$x^3 = 125\$; solve for x; \$x = 5\$

\$x^3 = -125\$; solve for x; \$x = -5\$

\$√x * √y = √{xy}\$

\$√{xy} = √x * √y\$

\$√x / √y = √{x/y}\$

\$√{x/y} = √x / √y\$

\$a√r + b√r = (a+b)√r\$

\$(a+b)√r = a√r + b√r\$

\$a√r - b√r = (a-b)√r\$

\$(a-b)√r = a√r - b√r\$

\$(√x)^n = √x^n\$

\$√x^n = (√x)^n\$

\$x^{1/n} = √^nx\$

\$√^nx = x^{1/n}\$

\$x^{n/m} = √^mx^n\$

\$√^mx^n = x^{n/m}\$

\$√a + √b ≠ √{a+b}\$

\$√4 = 2\$

\$√9 = 3\$

\$√16 = 4\$

\$√25 = 5\$

\$√36 = 6\$

\$√49 = 7\$

\$√64 = 8\$

\$√81 = 9\$

\$√100 = 10\$

\$√121 = 11\$

\$√144 = 12\$

\$√164 = 13\$

\$√196 = 14\$

\$√225 = 15\$

\$√256 = 16\$

\$√289 = 17\$

\$√324 = 18\$

\$√361 = 19\$

\$√400 = 20\$

\$√625 = 25\$

\$√900 = 30\$

\$√^3 8 = 2\$

\$√^3 27 = 3\$

\$√^3 64 = 4\$

\$√^3 125 = 5\$

\$√^3 216 = 6\$

\$√^3 344 = 7\$

\$√^3 512 = 8\$

\$√^3 729 = 9\$

\$√^3 1000 = 10\$

\$√^3 8000 = 20\$

\$2^2 = 4\$

\$3^2 = 9\$

\$4^2 = 16\$

\$5^2 = 25\$

\$6^2 = 36\$

\$7^2 = 49\$

\$8^2 = 64\$

\$9^2 = 81\$

\$10^2 = 100\$

\$11^2 = 121\$

\$12^2 = 144\$

\$13^2 = 164\$

\$14^2 = 196\$

\$15^2 = 225\$

\$16^2 = 256\$

\$17^2 = 289\$

\$18^2 = 324\$

\$19^2 = 361\$

\$20^2 = 400\$

\$25^2 = 625\$

\$30^2 = 900\$

\$2^3 = 8\$

\$3^3 = 27\$

\$4^3 = 64\$

\$5^3 = 125\$

\$6^3 = 216\$

\$7^3 = 344\$

\$8^3 = 512\$

\$9^3 = 729\$

\$10^3 = 1000\$

\$20^3 = 8000\$

compound interest = principal * (1 + (interest/C))time*C where C is the number of times compounded annually

Pythagorean Triplets:
3,4,5
5,12,13
8,15,17

Order of Operations:
PEDMAS - Parenthesis, Exponents, Multiplication and Division, Addition and Subtraction

\$√2≈1.41\$

\$√3≈1.73\$

\$√5≈2.24\$

\$√6≈2.45\$

\$√7≈2.65\$

\$√8≈2.83\$

\$√10≈3.16\$