GMAT Number Properties Questions

Number properties questions test what you know about integers: divisibility, factors, primes, odd and even behavior, remainders and exponent patterns. No outside formula sheet helps here, only the rules themselves. These questions appear throughout GMAT Quantitative Reasoning, which runs 21 Problem Solving questions in 45 minutes, and no calculator is allowed on this section. Speed matters here, and solid integer rules turn a long calculation into a quick lookup.

This guide covers divisibility and factor rules, prime numbers and prime factorization, odd and even behavior, positive and negative sign rules, remainder problems, and units-digit exponent patterns. Each section states the rule first, then shows the rule working on a real number. Three sample problems close out the guide, combining several rules the way GMAT questions do. Solve them with pen and paper before you read the walkthroughs.

Divisibility and Factor Rules

A divisibility rule lets you check whether one integer divides another without doing long division. Each rule reads the digits of a number instead of the full value, so the check takes seconds. Memorize the short list below and apply it straight to the answer choices. On a section with no calculator, this habit keeps you off long division.

Quick Divisibility Tests

The table below gives the test for each common divisor, plus a worked example of the test in use. Divisors 2 through 11 cover almost every divisibility question you will meet.

Finding All Factors of a Number

To list every factor of an integer, test divisors starting at 1 and stop once you pass the square root of the number. Each divisor pairs with a matching factor on the other side of the pair. Write down both members as you go, then sort the full set at the end. Stopping at the square root halves the work, because every pair above it repeats one you already found.

For 60, the square root sits near 7.7, so test 1 through 7. Divisors 1, 2, 3, 4, 5 and 6 each divide 60 evenly, pairing with 60, 30, 20, 15, 12 and 10. Divisor 7 fails. The full factor list is 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60, twelve factors in total.

Prime Numbers and Prime Factorization

A prime number has exactly two positive divisors: 1 and itself. This single rule filters out most integers you will see on the exam. Prime factorization takes the idea further by rewriting any integer as a product of primes. Once a number sits in prime form, questions about factors, multiples and divisibility turn into simple counting.

What Makes a Number Prime

The points below settle the edge cases test takers miss most often. Learn the first ten primes by heart so a quick check never slows you down.

• The number 1 has only one divisor, so it does not count as prime.
• 2 is the smallest prime number and the only even prime.
• The first several primes are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.
• Every composite number breaks down into a unique product of primes.

Breaking a Number into Prime Factors

Use a factor tree to break an integer into primes: divide by the smallest fitting prime, then repeat on the result. For 84, divide by 2 to get 42, divide 42 by 2 to get 21, then divide 21 by 3 to get 7, a prime. The prime factorization of 84 is 2 squared, times 3, times 7.

Prime factorization also gives you a fast way to count total factors. Write the number as primes raised to powers, add 1 to each exponent, then multiply the results together. For 60, equal to 2 squared times 3 times 5, the exponents are 2, 1 and 1, so the factor count is 3 times 2 times 2, or 12, matching the list above.

Odd, Even, Positive and Negative Rules

Odd and even rules and positive and negative rules answer a large share of number properties questions without any calculation at all. The exam asks whether a result has to be even, or whether a product ends up negative. You settle both with a rule instead of arithmetic. Learn the two short sets below and the answer arrives in a few seconds.

Odd and Even Arithmetic Rules

The table below shows the result of adding or multiplying odd and even integers. Subtraction follows the same pattern as addition.

Positive and Negative Multiplication Rules

Sign rules for multiplication and division reduce to the short list below. Count the negative factors and the sign of the product falls out at once.

• Positive times positive gives a positive result.
• Negative times negative gives a positive result.
• Positive times negative gives a negative result.
• An even count of negative factors multiplied together gives a positive product.
• An odd count of negative factors multiplied together gives a negative product.

Remainder Problems

Remainder questions rest on one formula and one habit: testing small numbers against the statement. Write the division out in full, then list values fitting the condition. Concrete numbers expose the pattern faster than abstract algebra on nearly every remainder problem. The two parts below give you the formula and the plug-in method.

The Remainder Formula

Dividend equals divisor times quotient, plus remainder. The remainder always sits at 0 or higher, and always stays smaller than the divisor. For 47 divided by 5, the quotient is 9 and the remainder is 2, since 5 times 9 plus 2 equals 47.

Testing Remainder Statements with Real Numbers

When a problem states a remainder condition, plug in real numbers instead of working with variables alone. Suppose n divided by 6 leaves a remainder of 4, and the question asks whether n is even. Try n equals 4: 4 divided by 6 leaves remainder 4, and 4 is even. Try n equals 10: 10 divided by 6 leaves remainder 4, and 10 is even.

Every value fitting this pattern equals a multiple of 6 plus 4, and a multiple of 6 is always even, so adding 4 keeps the sum even. This plug-in approach works well on Data Sufficiency questions built around remainders.

Units-Digit and Exponent Patterns

Large exponents look intimidating, but the units digit of a power always repeats in a short cycle. No base digit runs a cycle longer than four steps. Find the cycle, divide the exponent by its length, then read off the matching entry. The whole process takes well under a minute with no calculator.

Units-Digit Cycles by Base

The table below lists the units-digit cycle for each base digit and how many steps the cycle takes before it repeats.

Finding the Units Digit of a Large Power

Divide the exponent by the cycle length and look at the remainder. A remainder of 0 points to the last entry in the cycle, and any other remainder points to the matching position in the list. For 7 raised to the 38th power, the cycle for 7 is 7, 9, 3, 1, a length of 4. Dividing 38 by 4 gives a remainder of 2, so the units digit matches the second entry in the cycle, 9.

Sample Problem Walkthroughs

The three examples below combine rules from more than one section, the way most GMAT number properties questions do. Each problem is original, written for this guide. Solve each one on paper first, then compare your steps against the walkthrough underneath it. Where your method differs from the walkthrough matters more than matching the final answer.

Example 1. How many total factors does 180 have, and is 180 divisible by 11?

Break 180 into primes: 180 equals 2 squared, times 3 squared, times 5. Add 1 to each exponent and multiply: 3 times 3 times 2 equals 18 total factors. For divisibility by 11, take the alternating digit sum from right to left: 0 minus 8 plus 1 equals negative 7, not a multiple of 11, so 180 does not divide evenly by 11.

Example 2. A positive integer n leaves a remainder of 3 when divided by 5, and a remainder of 2 when divided by 4. What is the smallest value of n greater than 10?

List values fitting the first condition: 3, 8, 13, 18, 23, 28. List values fitting the second: 2, 6, 10, 14, 18, 22. The value 18 appears on both lists and sits above 10, and no smaller shared value does, so n equals 18.

Example 3. What is the units digit of 3 raised to the 47th power, plus 4 raised to the 47th power?

The cycle for 3 is 3, 9, 7, 1, a length of 4. Dividing 47 by 4 gives a remainder of 3, pointing to the third entry, 7, so 3 to the 47th power ends in 7. The cycle for 4 is 4, 6, a length of 2. Dividing 47 by 2 gives a remainder of 1, pointing to the first entry, 4, so 4 to the 47th power ends in 4. Adding the two units digits, 7 plus 4 equals 11, so the sum ends in 1.

Practice these rules with our free Number Properties drill, offering three difficulty levels with instant scoring and no signup required. For the full breakdown of the Quant Reasoning section, read our GMAT Quant Reasoning guide, and for the reasoning skills tested on remainder and divisibility statements, see our Data Sufficiency guide. Visit our GMAT prep page for the full library of mocks and section practice, or check your target score with our GMAT score calculator.

Sources

• GMAT Exam Content - mba.com
• GMAT Exam Structure - mba.com