GRE Quiz: Integer Properties Divisibility

15 questions · exam conditions

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Integer Properties DivisibilityQuestion 1 of 15

Integer $t$ is such that $t^2$ is divisible by $12$ . Which of the following must be true?

t

is divisible by

12

t

is divisible by

6

t

is divisible by

4

t

is even and divisible by

3

t

is odd and divisible by

3

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GRE Quiz

GRE Quiz: Integer Properties Divisibility

Practice Integer Properties Divisibility in GRE with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Integer Properties Divisibility, giving you a quick way to practice the rules, question types, and explanations that matter most for GRE.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Integer $t$ is such that $t^2$ is divisible by $12$ . Which of the following must be true?

$t$ is divisible by $12$
$t$ is divisible by $6$
$t$ is divisible by $4$
$t$ is even and divisible by $3$ (correct answer)
$t$ is odd and divisible by $3$

Explanation: This question tests integer properties and divisibility, examining conditions for a square to be divisible by 12. The key rule is that 12 = 4 * 3, so t^2 must be divisible by both 4 (requiring t even) and 3 (requiring t divisible by 3, as 3 is prime). Applying this, t even ensures t^2 ≡ 0 mod 4, and t divisible by 3 ensures t^2 ≡ 0 mod 3, combining to t^2 ≡ 0 mod 12. Choice D (t even and divisible by 3) must be true and satisfies all conditions, equivalent to t divisible by 6. For example, choice A assumes t divisible by 12, but t=6 is even and divisible by 3 with t^2=36 divisible by 12, yet 6 not divisible by 12. Similarly, choice E assumes t odd, but odd t yields odd t^2 not divisible by 4.

Question 2

Integers $a$ and $b$ satisfy $a\equiv 3\pmod{4}$ and $b\equiv 2\pmod{4}$ . Which of the following statements is true?

$a+b$ is divisible by $4$
$ab$ is divisible by $4$
$a-b$ is even
$a+b$ is odd (correct answer)
$ab$ is odd

Explanation: This question tests integer properties and divisibility, using modular arithmetic modulo 4. The key rule is that congruences determine parities: a ≡ 3 mod 4 means a odd, and b ≡ 2 mod 4 means b even. Applying this, a + b ≡ 3 + 2 ≡ 1 mod 4, which is odd, as odd + even = odd. No other choices are always true, such as a + b divisible by 4 (≡1 ≠0 mod 4). Thus, choice D (a + b is odd) is true and satisfies all conditions. For example, choice A assumes a + b divisible by 4, but ≡1 mod 4 contradicts this. Similarly, choice E assumes ab odd, but odd * even = even, failing the condition.

Question 3

Integers $m$ and $n$ satisfy $\gcd(m,n)=6$ and $\operatorname{lcm}(m,n)=180$ . Which of the following could be the value of $m$ ?

$12$
$18$ (correct answer)
$24$
$30$
$45$

Explanation: This question tests integer properties and divisibility, particularly the relationship between GCD and LCM. The key rule is that gcd(m, n) * lcm(m, n) = m * n, so here 6 * 180 = 1080 = m * n. Applying this, let m = 6a and n = 6b with gcd(a, b) = 1, then lcm(m, n) = 6 * a * b = 180, so a * b = 30. Possible pairs (a, b) with gcd 1 include (3, 10), giving m = 18 and n = 60, which satisfy gcd=6 and lcm=180. Thus, 18 could be the value of m and satisfies all conditions. For example, choice C (24) fails because 24 = 6*4, but 30/4 is not an integer, so no corresponding b with gcd(a, b)=1. Similarly, choice E (45) assumes a=7.5, which is not an integer.

Question 4

Integers $x$ and $y$ satisfy $x\equiv 1\pmod{6}$ and $y\equiv 5\pmod{6}$ . Which of the following must be true?

$x+y$ is divisible by $6$ (correct answer)
$xy$ is divisible by $6$
$x-y$ is divisible by $6$
$x+y$ is odd
$xy$ is even

Explanation: This question tests integer properties and divisibility, using congruences modulo 6. The key rule is that x ≡ 1 mod 6 and y ≡ 5 mod 6 imply x = 6p + 1 and y = 6q + 5. Applying this, x + y = 6(p + q + 1), which is divisible by 6. Other properties like xy even do not always hold, as x=1 and y=5 give odd xy=5. Thus, choice A (x + y divisible by 6) must be true and satisfies all conditions. For example, choice D assumes x + y odd, but it is always even, contradicting the assumption. Similarly, choice E assumes xy even, but it can be odd when both are odd.

Question 5

Integers $n$ and $k$ satisfy that $n$ is divisible by $4$ and $n+2k$ is odd. Which of the following must be true?

$k$ is even
$k$ is odd
$n$ is odd
$n+2k$ is divisible by $4$
No such integers $n$ and $k$ exist (correct answer)

Explanation: This question tests integer properties and divisibility, combining divisibility and parity. The key rule is that n divisible by 4 implies n even (actually n ≡ 0 mod 4), so n + 2k = 4m + 2k = 2(2m + k), which is even. Applying this, an even number cannot be odd, so no such n and k exist. Attempts to find examples always yield even n + 2k, contradicting the odd requirement. Thus, choice E (no such integers exist) must be true and satisfies all conditions. For example, choice A assumes k even, but even if k even, n + 2k remains even, failing the odd condition. Similarly, choice C assumes n odd, but n divisible by 4 cannot be odd.

Question 6

Integers $r$ and $s$ satisfy that $r$ is a multiple of $9$ and $s$ is a multiple of $6$ . Which of the following must be a multiple of $3$ ?

$r+s$
$r-s$
$rs$ (correct answer)
$\dfrac{r}{s}$
$r+2s$

Explanation: This question tests integer properties and divisibility, focusing on multiples and common factors. The key rule is that if r is a multiple of 9 and s of 6, then r ≡ 0 mod 9 and s ≡ 0 mod 6, implying both are multiples of 3. Applying this, rs = (9k)(6m) = 54km, which is divisible by 3 (actually by 54). Other expressions like r + s = 9k + 6m = 3(3k + 2m) are also multiples of 3, but the question specifies rs. Thus, choice C (rs) must be a multiple of 3 and satisfies all conditions. For example, choice D (r/s) assumes a ratio that may not be an integer, let alone a multiple of 3. Similarly, choice A (r + s) is a multiple of 3 but is not the only one.

Question 7

Integers $a$ and $b$ satisfy that $a\equiv 2\pmod{3}$ and $b\equiv 2\pmod{3}$ . Which of the following statements must be true?

$a+b$ is divisible by $3$
$ab$ is divisible by $3$
$a-b$ is divisible by $3$ (correct answer)
$a+b$ leaves remainder $2$ when divided by $3$
$ab$ leaves remainder $2$ when divided by $3$

Explanation: This question tests integer properties and divisibility, using congruences modulo 3. The key rule is that a ≡ 2 mod 3 and b ≡ 2 mod 3 imply a - b ≡ 0 mod 3, as both have the same remainder. Applying this, a - b = (3p + 2) - (3q + 2) = 3(p - q), divisible by 3. Other expressions like a + b ≡ 4 ≡ 1 mod 3 are not divisible by 3. Thus, choice C (a - b divisible by 3) must be true and satisfies all conditions. For example, choice A assumes a + b divisible by 3, but ≡1 mod 3 contradicts this. Similarly, choice D assumes remainder 2 for a + b, but it is actually 1 mod 3.

Question 8

Integers $a$ and $b$ satisfy that $a$ is a multiple of $4$ and $b$ is a multiple of $6$ . Which of the following statements must be true?

$a+b$ is a multiple of $12$
$ab$ is a multiple of $24$ (correct answer)
$a-b$ is a multiple of $6$
$\gcd(a,b)=2$
$\operatorname{lcm}(a,b)=12$

Explanation: This question tests integer properties and divisibility, focusing on multiples and their products. The key rule is that a multiple of 4 and b multiple of 6 imply a = 4p and b = 6q, so ab = 24pq, divisible by 24. Applying this, the product always includes factors 83=24 (from 46). Other statements like a + b multiple of 12 do not always hold, as 4+6=10 not divisible by 12. Thus, choice B (ab multiple of 24) must be true and satisfies all conditions. For example, choice A assumes a + b multiple of 12, but fails for a=4, b=6 due to invalid assumption about additivity. Similarly, choice D assumes gcd=2, but can be larger like 12 for a=12, b=12.

Question 9

Integers $a$ and $b$ satisfy that $a$ is odd and $ab$ is divisible by $8$ . Which of the following must be true?

$b$ is divisible by $8$ (correct answer)
$b$ is even
$b$ is divisible by $4$
$b$ is divisible by $2$ but not by $4$
$a$ is divisible by $8$

Explanation: This question tests integer properties and divisibility, focusing on powers of 2 when one factor is odd. The key rule is that if a is odd and ab divisible by 8=2^3, then b must provide all three factors of 2, so b divisible by 8. Applying this, if b not divisible by 8, ab would have fewer than three factors of 2. Examples like a=1 (odd), b=8 give ab=8 divisible by 8. Thus, choice A (b divisible by 8) must be true and satisfies all conditions. For example, choice B assumes b even (divisible by 2), but b=2 gives ab=2a divisible by 2 not 8, an invalid assumption about insufficient powers. Similarly, choice D assumes divisible by 2 but not 4, yielding ab divisible by 2 not 8.

Question 10

Integer $n$ is divisible by $15$ but not divisible by $45$ . Which of the following could be the value of $n$ ?

$45$
$-30$ (correct answer)
$90$
$-135$
$0$

Explanation: This question tests integer properties and divisibility, distinguishing multiples of 15 from those of 45. The key rule is that divisible by 15 means divisible by 3 and 5, but not by 45 means not divisible by 9 ( $3^2$ ). Applying this, -30 is divisible by 15 (30/15=2) but not by 45 (30/45=2/3 not integer), with exactly one factor of 3. Zero is divisible by both, failing the 'not' condition. Thus, -30 could be the value of n and satisfies all conditions. For example, choice A (45) is divisible by 45, violating the condition. Similarly, choice E (0) is divisible by 45, as 0=45*0.

Question 11

Integers $x$ and $y$ satisfy $x\mid 30$ and $y\mid 30$ . Which of the following could be the value of $\gcd(x,y)$ ?

$7$
$8$
$10$ (correct answer)
$12$
$25$

Explanation: This question tests integer properties and divisibility, particularly GCD of divisors of 30. The key rule is that if x and y both divide 30, their GCD must also divide 30. Applying this, possible GCDs are divisors of 30 like 1,2,3,5,6,10,15,30; 10 occurs for x=10, y=30 with gcd=10. Other choices like 7,8,12,25 do not divide 30. Thus, 10 could be the value of gcd(x,y) and satisfies all conditions. For example, choice A (7) fails because 7 does not divide 30, invalid for any divisors x and y. Similarly, choice E (25) assumes a divisor not factoring 30.

Question 12

Integer $n$ satisfies $-3\le n\le 3$ and $n\equiv 1\pmod{2}$ . Which of the following could be the value of $n$ ?

$0$
$2$
$-2$
$3$ (correct answer)
$4$

Explanation: This question tests integer properties and divisibility, combining inequalities with congruences. The key rule is that n ≡ 1 mod 2 means n is odd, and it must satisfy -3 ≤ n ≤ 3. Applying this, possible odd integers in the range are -3, -1, 1, 3. Among the choices, 3 is odd and within -3 ≤ 3 ≤ 3. Thus, 3 could be the value of n and satisfies all conditions. For example, choice A (0) is even, failing n ≡ 1 mod 2. Similarly, choice E (4) exceeds the upper bound of 3.

Question 13

Integers $n$ and $m$ satisfy $n\equiv -1\pmod{7}$ and $m\equiv 3\pmod{7}$ . Which of the following is the remainder when $nm$ is divided by $7$ ?

$1$
$2$
$3$
$4$ (correct answer)
$6$

Explanation: This question tests integer properties and divisibility, specifically multiplication in modular arithmetic. The key rule is that congruences multiply: n ≡ -1 mod 7 and m ≡ 3 mod 7 imply nm ≡ (-1)*3 ≡ -3 ≡ 4 mod 7 (since -3 + 7 = 4). Applying this, the remainder is always 4 for any such n and m. No other remainder is possible under these congruences. Thus, 4 is the remainder and satisfies all conditions. For example, choice A (1) assumes a different product, but -3 ≡ 4 mod 7, not 1. Similarly, choice E (6) fails as it assumes ≡ -1 mod 7, but the product is ≡ 4.

Question 14

Integers $x$ and $y$ satisfy that $x+y$ is odd and $xy$ is even. Which of the following statements must be true?

$x$ and $y$ are both even
$x$ and $y$ are both odd
Exactly one of $x$ and $y$ is even (correct answer)
$x-y$ is odd
$x+y$ is divisible by $4$

Explanation: This question tests integer properties and divisibility, focusing on parity and even-odd relationships. The key rule is that the sum of two integers is odd if and only if one is even and one is odd, while their product is even if at least one is even. Applying this, x + y odd implies exactly one of x or y is even, which also ensures xy is even since at least one factor is even. No other configuration satisfies both conditions simultaneously, as both even or both odd would make x + y even. Therefore, choice C (exactly one of x and y is even) must be true and satisfies all conditions. For example, choice A assumes both even, but this would make x + y even, contradicting the given odd sum. Similarly, choice B assumes both odd, which also results in an even sum, failing the condition.

Question 15

Integers $a$ and $b$ satisfy $a\mid b$ and $b\mid a$ . Which of the following must be true?

$a=b$
$a=-b$
$|a|=|b|$ (correct answer)
$a$ and $b$ are both positive
$a$ and $b$ are both even

Explanation: This question tests integer properties and divisibility, specifically the condition where two integers divide each other mutually. The key rule is that if a divides b and b divides a, then a and b must have the same absolute value, as they are either equal or negatives of each other. Applying this rule, a divides b implies b = k * a for some integer k, and b divides a implies a = m * b for some integer m, leading to a = m * k * a and thus m * k = 1. The integer solutions to m * k = 1 are (1,1) and (-1,-1), corresponding to b = a or b = -a. Therefore, |a| = |b| must be true and satisfies all conditions by encompassing both possibilities, including the case where a = b = 0. For example, choice A assumes a = b, but this fails when a = 2 and b = -2, as they divide each other but are not equal. Similarly, choice B assumes a = -b, but this fails when a = b = 3.