Is 1 a prime number on the GRE?

No. A prime number has exactly two positive divisors, 1 and itself, so 1 is not prime because it has only one divisor. The smallest prime is 2, which is also the only even prime.

How do you find how many factors a number has?

Write the prime factorization, add 1 to each exponent, and multiply the results. For example, 72 = 2³ × 3², so it has (3 + 1)(2 + 1) = 12 positive divisors.

What is the relationship between GCF and LCM?

For any two positive integers a and b, GCF(a, b) × LCM(a, b) = a × b. This lets you recover a missing number when you know the GCF, the LCM, and one of the two integers.

How do I handle remainders when a number is multiplied or squared?

Replace the number with its remainder, perform the operation, then reduce the result back into the range 0 to d − 1. For instance, if n leaves remainder 4 mod 7, then 3n leaves the remainder of 12 mod 7, which is 5.

CramClub • Number Properties and Divisibility — GRE® General Test Guide & Practice

even $\pm$ even $=$ even; odd $\pm$ odd $=$ even; even $\times$ any $=$ even.

If $n = p_1^{a_1} p_2^{a_2}\cdots$ , then $d(n) = (a_1 + 1)(a_2 + 1)\cdots$

$\text{GCF}(a,b) \times \text{LCM}(a,b) = a \times b$

$n = dq + r$ , with $0 \le r < d$

A product of $k$ consecutive integers is divisible by $k!$

Integers Are the GRE's Favorite Playground

Number-property questions rarely ask you to calculate much. Instead they hand you a fact about an integer — it is even, it is prime, it leaves a remainder of $3$ when divided by $5$ — and ask what must be true, what could be true, or what cannot be true. The reward goes to test-takers who reason from definitions and stay alert to the two integers that break patterns: $0$ and $1$ , and to negatives. Almost every wrong answer on these items comes from a hidden assumption that a variable is positive, or an integer, or distinct from another.

Factors, Multiples, and Primes

A factor (or divisor) of an integer $n$ divides it with no remainder; a multiple of $n$ is any product $n \cdot k$ for integer $k$ . Every integer greater than $1$ is either prime (exactly two positive divisors, $1$ and itself) or composite. Memorize the small primes — $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$ — and remember the traps: $1$ is not prime, and $2$ is the only even prime. The prime factorization of an integer is unique, and it is the master key: from $n = p_1^{a_1} p_2^{a_2}\cdots$ you can read off the count of divisors, the GCF, and the LCM.

Parity: Even and Odd

Parity questions collapse to a few rules that never fail:

Operation	Result
even $\pm$ even, odd $\pm$ odd	even
even $\pm$ odd	odd
even $\times$ any integer	even
odd $\times$ odd	odd

There is no reliable rule for division, because the quotient need not be an integer. A useful consequence: the sum of two integers is even exactly when they share the same parity.

Divisibility Tests

You can decide divisibility without long division:

2: last digit is even.
3: digit sum is divisible by $3$ .
4: the number formed by the last two digits is divisible by $4$ .
5: last digit is $0$ or $5$ .
6: divisible by both $2$ and $3$ .
9: digit sum is divisible by $9$ .
8: last three digits are divisible by $8$ .

Remainders

When $n$ is divided by $d$ , we can write $n = dq + r$ with $0 \le r < d$ . The GRE loves remainder chains: if $n$ leaves remainder $4$ on division by $7$ , then $n = 7q + 4$ , so $3n = 21q + 12 = 7(3q + 1) + 5$ — the remainder of $3n$ is $5$ , not $12$ . Reduce the remainder back into the range $0 \le r < d$ at every step. Cyclic behavior (units digits of powers, days of the week) is just remainder arithmetic in disguise.

GCF and LCM

The greatest common factor (GCF) is the product of the shared prime powers at their lowest exponent; the least common multiple (LCM) uses the highest exponent. The identity that saves time:

$\text{GCF}(a, b) \times \text{LCM}(a, b) = a \times b.$

So if two numbers have GCF $6$ and LCM $90$ , their product is $540$ ; given one number is $18$ , the other is $540 \div 18 = 30$ .

Counting Divisors

From $n = p_1^{a_1} p_2^{a_2}\cdots$ , the number of positive divisors is

$(a_1 + 1)(a_2 + 1)\cdots$

For $72 = 2^3 \cdot 3^2$ , that is $(3+1)(2+1) = 12$ divisors. This one formula answers a whole family of "how many factors" questions instantly.

Consecutive Integers

Consecutive integers hide clean structure. Among any $k$ consecutive integers, exactly one is divisible by $k$ , so the product of $k$ consecutive integers is always divisible by $k!$ — three in a row is always divisible by $6$ . The sum of an odd count of consecutive integers is divisible by that count (the middle term is the average). These facts turn intimidating "must be divisible by" questions into one-line arguments.

A Reliable Game Plan

"Must be true"? Try to break the statement with $0$ , $1$ , a negative, or a non-integer before trusting it.
Divisibility? Use the digit tests; for $6$ , $12$ , or $15$ , split into coprime factors and test each.
Remainders? Write $n = dq + r$ and reduce back into range after every operation.
Factors/GCF/LCM? Prime-factorize first; everything else follows.

If $a$ is odd and $b$ is even, is $3a^2 + 2b - 1$ even or odd?

The four-digit number $5{,}2d4$ is divisible by $3$ . What is the smallest possible digit $d$ ?

How many positive divisors does $360$ have?

Two positive integers have GCF $8$ and LCM $240$ . One of them is $48$ . What is the other?

When the positive integer $n$ is divided by $5$ , the remainder is $3$ . What is the remainder when $n^2 + 2n$ is divided by $5$ ?

If $n$ is a positive integer, prove that $n^3 - n$ is always divisible by $6$ .

Assuming an unknown is a positive integer. On "must be true" items, a variable can be $0$ , $1$ , negative, or a fraction unless the problem says otherwise.

Actively try to break the claim with $0$ , $1$ , and a negative before accepting it. One counterexample kills a "must" answer.

Calling $1$ prime, or forgetting that $2$ is prime. Both mistakes corrupt factor counts and "how many primes" questions.

Prime means exactly two positive divisors: $1$ is not prime, and $2$ is the only even prime.

Leaving a remainder out of range, e.g. reporting the remainder of $3n$ as $12$ when dividing by $7$ .

A remainder must satisfy $0 \le r < d$ . After any operation, subtract multiples of $d$ until the remainder is back in range.

Multiplying two numbers to get their LCM, or confusing GCF with LCM in the identity.

GCF uses the lowest shared prime powers, LCM the highest; only their product equals $a \times b$ .

When the positive integer $n$ is divided by $7$ , the remainder is $4$ . What is the remainder when $3n$ is divided by $7$ ?

Two positive integers have greatest common factor $6$ and least common multiple $90$ . If one of the integers is $18$ , what is the other integer?

How many positive integers less than $100$ are divisible by neither $2$ nor $3$ ?

$n$ is a positive integer.

Quantity A: the remainder when $n^2$ is divided by $4$

Quantity B: $1$

The sum of five consecutive integers is $5$ more than the sum of the first three of those integers. What is the greatest of the five integers?

If $p$ and $q$ are distinct prime numbers, each greater than $2$ , which of the following must be true?

I. $p + q$ is even

II. $pq$ is odd

III. $p + q + 1$ is prime