GRE General Test

Integers and signs

The integers are the whole numbers and their negatives: $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$ Positive integers lie above 0 and negative integers below it, and 0 itself is neither. Adding, subtracting or multiplying integers always gives an integer; dividing one integer by another may not.

The sign of a product follows three facts. A positive times a positive is positive, a negative times a negative is positive, and a positive times a negative is negative. So $(-4)(-6) = 24$ and $(-4)(6) = -24$. With more than two factors, the product is negative when an odd number of the factors are negative and positive when an even number are.

Factors and multiples

When integers are multiplied, each of them is a factor (or divisor) of the product. Since $84 = 7 \times 12$, both 7 and 12 are factors of 84. The positive factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84, and the negatives of these are factors as well. Factors come in pairs that multiply to 84, which is a reliable way to check a list: $1 \times 84$, $2 \times 42$, $3 \times 28$, $4 \times 21$, $6 \times 14$, $7 \times 12$.

A number is a multiple of each of its factors, and is divisible by them. A few facts follow directly.

• Every nonzero integer has infinitely many multiples: 25, 50, 75, and so on.
• 1 is a factor of every integer, and 0 is a multiple of every integer.
• 0 is a factor of no integer except 0, because nothing multiplied by 0 gives a nonzero number.

Divisibility tests

Some divisibility can be read off the digits.

For 6,123 the digits add to 12, so it is divisible by 3 but not by 9. For 7,216 the last three digits give $216 = 8 \times 27$, so it is divisible by 8.

Least common multiple and greatest common divisor

The least common multiple of two nonzero integers is the smallest positive integer that is a multiple of both. The greatest common divisor (or greatest common factor) is the largest positive integer that divides both. The quickest way to find either is through prime factorisations.

Take 48 and 180. Their factorisations are $48 = 2^4 \times 3$ and $180 = 2^2 \times 3^2 \times 5$.

• The greatest common divisor uses each prime at the lower of its two powers: $2^2 \times 3 = 12$.
• The least common multiple uses each prime at the higher of its two powers: $2^4 \times 3^2 \times 5 = 720$.

The product of the two numbers equals the product of their greatest common divisor and least common multiple, which gives a check: $48 \times 180 = 8{,}640$ and $12 \times 720 = 8{,}640$.

Quotient and remainder

When an integer $c$ is divided by a positive integer $d$, find the largest multiple of $d$ that is less than or equal to $c$. Write it as $qd$; then $q$ is the quotient and the remainder is $r = c - qd$, so that

The remainder is always at least 0 and less than $d$, and it is 0 exactly when $d$ divides $c$.

• $53$ divided by $8$: the largest multiple of 8 not above 53 is 48, so the quotient is 6 and the remainder is 5, since $53 = 6(8) + 5$.
• $9$ divided by $20$: the largest multiple of 20 not above 9 is 0, so the quotient is 0 and the remainder is 9.
• $-17$ divided by $5$: the largest multiple of 5 not above $-17$ is $-20$, so the quotient is $-4$ and the remainder is 3, since $-17 = (-4)(5) + 3$. The remainder is never negative, which is why the quotient is $-4$ and not $-3$.

Finding a number from its remainders

To find the least positive integer that leaves remainder 3 when divided by 4 and remainder 2 when divided by 5, list the numbers of the first kind: 3, 7, 11, 15, 19, and so on. The first one that leaves remainder 2 on division by 5 is 7, since $7 = 1(5) + 2$. So the answer is 7. Later solutions are 20 more each time, because 20 is the least common multiple of 4 and 5.

Even and odd integers

An integer divisible by 2 is even and any other integer is odd; an odd integer leaves remainder 1 when divided by 2. Six results cover sums and products.

Parity questions can be answered by applying these one step at a time. If $m$ is even and $n$ is odd, then $3m + 2n$ is even plus even, which is even, while $m + n^2$ is even plus odd (since $n^2$ is odd times odd), which is odd.

Prime numbers and prime factorisation

A prime number is an integer greater than 1 whose only positive divisors are 1 and itself. Below 10 the primes are 2, 3, 5 and 7, and the next six are 11, 13, 17, 19, 23 and 29. The number 1 is not prime, and 2 is the only even prime. An integer greater than 1 that is not prime is composite.

Every integer greater than 1 is either prime or can be written as a product of primes in exactly one way, apart from the order of the factors. That product is its prime factorisation. To find it, divide repeatedly by the smallest prime that goes in.

The prime divisors of 360 are therefore 2, 3 and 5. The factorisation also counts the positive divisors: add 1 to each exponent and multiply, giving $(3+1)(2+1)(1+1) = 24$ positive divisors of 360.

To test whether a number such as 91 is prime, try dividing by the primes up to its square root. Here $\sqrt{91}$ is a little under 10, so only 2, 3, 5 and 7 need checking, and $91 = 7 \times 13$ shows it is composite.