Number Properties, Fractions, and Percents

What GRE Arithmetic Actually Tests

GRE Arithmetic is never about raw computation — an on-screen calculator handles that. It rewards structural insight into how numbers behave: divisibility, remainders, factor relationships, and the interplay of fractions, decimals, and percents. Master the properties below and you can often answer without computing at all.

Integer Properties

Property	Rule	Why it matters on the GRE
Divisibility	An integer is divisible by 3 iff its digit sum is; by 4 iff its last two digits are; by 6 iff by both 2 and 3	Lets you eliminate answer choices instantly
Remainders	If $n = qd + r$, then $0 \le r < d$	Remainder questions are pure structure, not arithmetic
Prime factorization	Every integer > 1 factors uniquely into primes	Underlies GCF, LCM, and "how many factors" questions
GCF and LCM	For any two positive integers, $\text{GCF} \times \text{LCM} = \text{product}$	Converts one unknown into the other
Even/odd	even ± even = even; odd + odd = even; odd × odd = odd	Determines parity without specific values

Remainder shortcut. If $n$ leaves remainder 4 when divided by 7, then $n \equiv 4 \pmod 7$, and any expression in $n$ can be reduced mod 7. For example, $3n + 2 \equiv 3(4) + 2 = 14 \equiv 0 \pmod 7$ — the remainder is 0, found with no knowledge of $n$ itself.

Fractions, Decimals, and Percents Are One Idea

A percent is just a fraction over 100 and a decimal is just a fraction with a power-of-10 denominator. Fluency means converting whichever form makes the arithmetic cleanest.

Fraction	Decimal	Percent
$\frac{1}{8}$	0.125	12.5%
$\frac{1}{6}$	0.1666…	16.67%
$\frac{3}{8}$	0.375	37.5%
$\frac{5}{6}$	0.8333…	83.33%

"Percent of" translation: "What is 30% of 80?" → $0.30 \times 80 = 24$. "15 is what percent of 60?" → $\frac{15}{60} = 0.25 = 25\%$.

Percent Change — and the Successive-Change Trap

$\text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%$

The GRE's favorite trap: successive percent changes do not add. A 25% increase followed by a 20% decrease is not a net 5% increase. Multiply the factors:

$1.25 \times 0.80 = 1.00 \quad\Rightarrow\quad \text{net change } = 0\%.$

Ratios and Proportions

A ratio $a : b$ means the quantities are $ak$ and $bk$ for some multiplier $k$. Introduce $k$ and the problem becomes algebra. If A : B = 3 : 2 and you add 10 units of B to reach A : B = 3 : 4, write $A = 3k$, $B = 2k$, then $\frac{3k}{2k + 10} = \frac{3}{4}$ and solve for $k$.

Exponents and Roots

Rule	Statement
Product	$a^m \cdot a^n = a^{m+n}$
Quotient	$\dfrac{a^m}{a^n} = a^{m-n}$
Power of a power	$(a^m)^n = a^{mn}$
Negative exponent	$a^{-n} = \dfrac{1}{a^n}$
Fractional exponent	$a^{m/n} = \sqrt[n]{a^m}$
Radical sum	$\sqrt{a} + \sqrt{b} \ne \sqrt{a+b}$ — simplify each radical first

Common-base tactic: To compare $2^{40}$ with $4^{20}$, rewrite $4^{20} = (2^2)^{20} = 2^{40}$ — equal. Rewriting to a common base turns intimidating exponent comparisons into trivial ones.