If n is an integer that is divisible by 15, which of the following statements MUST also be… — GRE General Test

If $n$ is an integer that is divisible by $15$ , which of the following statements MUST also be true? Select all such statements.

$n$ is divisible by $3$ Cevap
$n$ is divisible by $5$ Cevap
$n + 15$ is divisible by $15$ Cevap
D
$n$ is divisible by $30$
E
$n$ is an even integer

Cevap

The statements asserting that

n

is divisible by

3

n

is divisible by

5

, and

n + 15

is divisible by

15

must all be true.

Any integer divisible by

15

must be divisible by all factors of

15

, which includes

3

and

5

. Furthermore, adding

15

to a multiple of

15

yields another multiple of

15

, so

n + 15

is also divisible by

15

Adım Adım Çözüm

Analyze the prime factor decomposition of the divisor

The prime factorization of

15

3 \times 5

. Therefore, any integer

n

that is a multiple of

15

can be expressed as

n = 15k = 3 \times 5 \times k

for some integer

k

Understanding factor relationships allows evaluation of divisibility rules.

Evaluate divisibility of

n

3

and

5

Since

n = 3(5k)

n

is divisible by

3

. Since

n = 5(3k)

n

is divisible by

5

. Both statements are always true.

Any multiple of a composite number is also a multiple of that number's factors.

Evaluate the expression

n + 15

n + 15 = 15k + 15 = 15(k + 1)

. Since

k + 1

is an integer,

n + 15

is a multiple of

15

. This statement is always true.

Adding a multiple of

15

to another multiple of

15

yields a multiple of

15

Test counterexamples for remaining statements

For

n = 15

15

is not divisible by

30

, so divisibility by

30

is not guaranteed. Additionally,

15

is odd, so being an even integer is not guaranteed.

A statement must hold for all possible values of

n

to be necessarily true.

Anahtar Kavram

Divisibility by a composite number implies divisibility by all of its factors, and multiples of a number remain multiples when another multiple of that number is added.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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