GMAT

Number property questions test how integers behave rather than how to compute with them: divisibility, primes, factors, remainders, units digits, and the effect of signs and parity. They often contain an unknown that cannot be solved for, and the answer depends on what must be true for every allowed value.

Integers and Types of Numbers

Term	Meaning	Notes
Integer	−3 −2 −1 0 1 2 3 (no fractional part) …, , , , , , , , …	Zeroisanintegerandiseven
Positive/negative	Greaterthanzero/lessthanzero	Zeroisneitherpositivenornegative
Factor(divisor)	disafactorofnifn÷disaninteger	Everyintegerhas1anditselfas
		factors
Multiple	nisamultipleofdifn=dkforanintegerk	0isamultipleofeveryinteger
Prime Positiveintegerwithexactlytwopositivefactors 1anditself 2istheonlyevenprime;1isnotprime , 1 Composite Integergreaterthan1thatisnotprime Hasafactorotherthan1anditself heprimeslessthan50are2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 Totestwhetheranumberisprime check , , , , , , , , , , , , , , . , ivisibilityonlybyprimesuptoitssquareroot:143isnotprimebecause143=11×13 and11islessthan√143≈12 , .

The primes less than 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. To test whether a number is prime, check divisibility only by primes up to its square root: 143 is not prime because 143 = 11 × 13, and 11 is less than √143 ≈ 12.

Divisibility Rules

Divisor	An integer is divisible if …	Example
2	itslastdigitiseven	1358 ,
3	thesumofitsdigitsisdivisibleby3	4572:4+5+7+2=18 ,
4	itslasttwodigitsformamultipleof4	7316:16=4×4 ,
5	itslastdigitis0or5	2045 ,
6	itisdivisiblebyboth2and3	3114:even digitsum9 , ,
8	itslastthreedigitsformamultipleof8	9136:136=8×17 ,
9	thesumofitsdigitsisdivisibleby9	6381:6+3+8+1=18 ,
10	itslastdigitis0	5290 ,
11	thealternatingsumofitsdigitsisdivisibleby11	9 ,284: 9 − 2 + 8 − 4 = 11
12	itisdivisiblebyboth3and4	5112:digitsum9;12=4×3 ,

Divisibility combines through prime factors: a number divisible by both 4 and 3 is divisible by 12, but a number divisible by 4 and 6 is only guaranteed divisible by LCM(4, 6) = 12, not by 24. If a divides b and b divides c, then a divides c; if d divides both x and y, then d divides x + y and x − y.

Prime Factorization and the Number of Factors

Every integer greater than 1 can be written as a product of primes in exactly one way. A factor tree builds the factorization by splitting any composite number into two factors until every branch ends in a prime.

If n = p^a × q^b × r^c (distinct primes), then

number of positive factors = (a + 1)(b + 1)(c + 1)

360 = 2³ × 3² × 5¹ has (3 + 1)(2 + 1)(1 + 1) = 24 factors. Each factor chooses an exponent of 2 from 0 to 3, of 3 from 0 to 2, and of 5 from 0 to 1. A perfect square has only even exponents, so it has an odd number of factors.

Squares and cubes: n is a perfect square exactly when every prime exponent is even (36 = 2² × 3²), and a perfect cube when every exponent is a multiple of 3.

Must-divide questions: if n² is divisible by 72 = 2³ × 3², then n² contains at least 2⁴ × 3² (exponents in a square are even), so n contains at least 2² × 3 and must be divisible by 12.

Greatest Common Divisor and Least Common Multiple

The greatest common divisor (GCD, or greatest common factor) of two integers takes each shared prime to its lowest power. The least common multiple (LCM) takes every prime that appears to its highest power. For any two positive integers, GCD × LCM equals the product of the numbers.

72 = 2³ × 3²

120 = 2³ × 3 × 5

GCD = 2³ × 3 = 24 (shared primes, lowest powers)

LCM = 2³ × 3² × 5 = 360 (all primes, highest powers)

Check: 24 × 360 = 8,640 = 72 × 120

Use the GCD for the largest equal-size groups or tiles (72 red and 120 blue beads split into identical kits: at most 24 kits). Use the LCM for events that repeat together (buses every 12 and 18 minutes leave together again after LCM(12, 18) = 36 minutes).

Remainders

When a positive integer n is divided by a positive integer d, the quotient q and remainder r satisfy n = qd + r, with 0 ≤ r < d. The remainder is always smaller than the divisor, and a remainder of 0 means d divides n. A statement such as "n leaves remainder 3 when divided by 7" describes a list: n = 7q + 3 = 3, 10, 17, 24, …. Writing out the first few values is often the fastest solution.

Fact	Rule	Example
Sumsanddifferences	Addorsubtractremainders thenreducebythe ,	aleaves4andbleaves5whendividedby
	divisor	7:a+bleaves9−7=2
Productsandpowers	Multiplyremainders thenreduce ,	nleaves4whendividedby6:n²leaves16
		4; 3n leaves 12 0 → →
Decimalform	n/d = q + r/d , so the decimal part equals r/d	If x/y = 12 .35 and the remainder is 7 ,1 then 7/y = 0 .35 and y = 20

Fact	Rule	Example
Remainderlargerthandividend	Ifn<d thequotientis0andtheremainderisn ,	5dividedby8leavesremainder5
Twoconditions	Listvaluesforthelargerdivisorandtestthem	Remainder3by5and4by7:test4 11 18 , ,
	againsttheother	→ n = 18 + 35k

Units Digits

The units digit of a sum or product depends only on the units digits of the numbers involved: the units digit of 347 × 1,289 is the units digit of 7 × 9 = 63, which is 3. The units digits of successive powers repeat in cycles of length 1, 2, or 4, so a large exponent reduces to its remainder when divided by 4.

Unitsdigit	Cycleofpowers	Length	Example
0 1 5 6 , , ,	Alwaysitself	1	6⁴⁴endsin6
4	4 6 ,	2	4toanoddpowerendsin4;evenpowerendsin
			6
9	9 1 ,	2	19²⁰endsin1
2	2 4 8 6 , , ,	4	2⁴²endsin4(42÷4leaves2)
3	3 9 7 1 , , ,	4	3²⁴endsin1(remainder0)
7	7 9 3 1 , , ,	4	7⁸³endsin3(remainder3)
8	8 4 2 6 , , ,	4	8⁵endsin8(remainder1)

The same cycling idea finds remainders for other divisors. Powers of 2 divided by 7 leave remainders 2, 4, 1, 2, 4, 1, …, a cycle of length 3; since 100 ÷ 3 leaves remainder 1, 2¹⁰⁰ divided by 7 leaves remainder 2.

Odd, Even, Positive, and Negative

Operation	Parityrule	Signrule
Addition	E+E=E;O+O=E;E+O=O	Positive+positiveispositive;mixedsigns
		dependonsizes
Subtraction	Sameasaddition:thedifferenceoftwooddsis	a − b is positive exactly when a > b
	even
Multiplication	E×anything=E;O×O=O	Negativewhenanoddnumberoffactorsare
		negative
Powers	OⁿisoddandEⁿisevenforpositiven	Evenpowersofnonzeronumbersarepositive
Division	Noparityrule:6/2=3isodd 8/2=4iseven 6/4is , ,	Samesignruleasmultiplication
	notaninteger

A product of integers is odd only if every factor is odd. The product of any two consecutive integers, n(n + 1), is always even. If xy < 0, then x and y have opposite signs; if x²y > 0, then y > 0 (and x ≠ 0), but x can have either sign.

Consecutive Integers

Evenly spaced sets: the mean equals the median equals (first + last)/2, and the sum equals the number of terms times the mean.

Counting terms: from a to b inclusive with spacing s, there are (b − a)/s + 1 terms. The even integers from 20 to 60 inclusive number (60 − 20)/2 + 1 = 21, with mean 40 and sum 21 × 40 = 840.

Sum of the first n positive integers: 1 + 2 + … + n = n(n + 1)/2, so 1 + 2 + … + 100 = 5,050.

Divisibility: any n consecutive integers include exactly one multiple of n, and their product is divisible by n!. The product of three consecutive integers is divisible by 6.

Counting multiples in a range: the multiples of 7 from 100 to 200 number ⌊200/7⌋ − ⌊99/7⌋ = 28 − 14 = 14.

Sums of consecutive integers: the sum of an odd number of consecutive integers is divisible by the number of terms; the sum of an even number of them is not.

Terminating Decimals and Trailing Zeros

A fraction in lowest terms has a terminating decimal exactly when its denominator has no prime factors other than 2 and 5. So 7/40 terminates (40 = 2³ × 5), 7/30 repeats (30 contains a 3), and 21/30 terminates because it reduces to 7/10. Always simplify before applying the test.

Each trailing zero needs one factor of 10 = 2 × 5; the 5s are scarcer.

zeros at the end of n! = ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + …

30! → 6 + 1 = 7 zeros

100! → 20 + 4 = 24 zeros

The same method counts any prime in a factorial: the power of 2 in 20! is ⌊20/2⌋ + ⌊20/4⌋ + ⌊20/8⌋ + ⌊20/16⌋ = 10 + 5 + 2 + 1 = 18.

When a question asks what must be true, test the allowed values that behave differently: 0, 1, a negative number, a fraction, an even and an odd integer. One counterexample eliminates a choice. When a question asks what could be true, one working example confirms a choice.

0 is even; 1 is not prime; 2 is the only even prime.

Units digits of powers cycle; reduce the exponent by its remainder when divided by 4.

Number of factors of p^a q^b r^c is (a + 1)(b + 1)(c + 1).

A product is odd only if every factor is odd; count negative

GCD: shared primes, lowest powers. LCM: all primes, highest

factors to find the sign.

powers. GCD × LCM = product.

A reduced fraction terminates only if its denominator has no

n = qd + r with 0 ≤ r < d; list values to solve remainder

primes other than 2 and 5.

problems.

Trailing zeros of n! = ⌊n/5⌋ + ⌊n/25⌋ + ….