🔢 Quant

Integers, Factors & Remainders

Primes, prime factorization, divisibility tests, remainders, parity and digit logic — the number-property toolkit Quant leans on.

~5h

to master

What the GMAT does with integers

Integer questions on the GMAT are never "recite the definition" — they hand you a property (divisibility, parity, a remainder) and ask you to use it: simplify, count, or pin down an unknown. Master three machines: prime factorization, the remainder identity, and parity rules.

Core rules

📐Core Rule

Divisibility lives in prime factorizations. Integer $a$ divides integer $b$ exactly when every prime in $a$ 's factorization appears in $b$ 's at least as high a power. $2376 = 2^3 \cdot 3^3 \cdot 11$ is divisible by $24 = 2^3 \cdot 3$ , but not by $16 = 2^4$ (not enough 2s) and not by $21 = 3 \cdot 7$ (no 7 at all).

Primes: exactly two positive divisors. $2$ is the only even prime; $1$ is not prime.
The remainder identity: dividing $n$ by $d$ gives unique $q, r$ with $n = dq + r$ and $0 \le r < d$ . When the dividend is smaller than the divisor, the quotient is $0$ and the whole dividend carries over as the remainder.
Remainders respect + and ×: the remainder of a sum (or product) equals the remainder of the sum (or product) of the remainders. To find $(143 + 31) \bmod 13$ : $143 \to 0$ , $31 \to 5$ , so the answer is $5$ — no big addition needed.
Parity: a product is even if any factor is even; a sum of two integers is even exactly when the two have the same parity. Consecutive integers always contain a multiple of 2, of 3 (among any three), and so on.

Divisibility tests (memorize cold): by 2 — last digit even · by 3 — digit sum divisible by 3 · by 4 — last two digits · by 5 — ends in 0/5 · by 9 — digit sum · by 10 — ends in 0.

✏️Worked Example

How many three-digit multiples of 7 are there? Smallest: $105 = 7 \cdot 15$ . Largest: $994 = 7 \cdot 142$ . Count $= 142 - 15 + 1 = 128$ . The $+1$ is the classic fence-post step — count posts, not gaps.

Last-digit arithmetic

The final digit of a sum or product depends only on the final digits of the inputs. The last digit of $345 \times 789$ is the last digit of $5 \times 9 = 45$ , i.e. $5$ . Powers cycle: $7^1, 7^2, 7^3, 7^4$ end in $7, 9, 3, 1$ and then repeat with period 4 — so the units digit of $7^{50}$ is the digit at position $50 \bmod 4 = 2$ in the cycle: $9$ .

Digits as algebra

A two-digit number with tens digit

t

and units digit

u

10t + u

; its reversal is

10u + t

. Their difference is

9(t - u)

— always a multiple of 9. Place-value translation turns digit puzzles into one-line algebra. Try it inline: .

⚠️GMAT Trap

"Divisible by 6 and 4" does not mean divisible by 24. Combine conditions with the LCM, not the product: $\text{lcm}(6,4) = 12$ (their prime demands overlap in a 2). The product rule only works when the divisors are coprime.

💡Exam Tip

When a question says "

n

leaves remainder 3 when divided by 7", immediately write

n = 7k + 3

and work with that expression. Concrete test values (

3, 10, 17

) are the fastest way to kill wrong options. Practice the move on .

Checklist

Factor into primes before reasoning about divisibility
Translate remainder statements into $n = dq + r$ form
Use parity to eliminate options before computing
Digit puzzles → place-value algebra ( $10t + u$ )
Count inclusive ranges with the $+1$

Sample Questions

22 practice questions

Medium

When the positive integer $n$ is divided by 9, the remainder is 5. What is the remainder when $3n + 8$ is divided by 9?

Medium

When the digits of a two-digit positive integer are reversed, the resulting integer is 36 greater than the original integer. What is the positive difference between the two digits?

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