How many positive integers n less than 30 satisfy the condition that n^2 - 1 is divisible … — GRE General Test

How many positive integers $n$ less than $30$ satisfy the condition that $n^2 - 1$ is divisible by $24$ ?

A
5
B
8
C
9
10Cevap
E
15

Cevap

For

n^2 - 1

to be divisible by

24 = 8 \times 3

n

must be an odd integer that is not divisible by 3, which means

\gcd(n, 6) = 1

. The positive integers less than 30 coprime to 6 are 1, 5, 7, 11, 13, 17, 19, 23, 25, and 29, totaling 10 integers.

Adım Adım Çözüm

Factor the expression and analyze divisibility requirements for 24.

n^2 - 1 = (n - 1)(n + 1)

. Since

24 = 8 \times 3

with

\gcd(8, 3) = 1

n^2 - 1

must be divisible by both

8

and

3

Decompose 24 into coprime prime-power factors to evaluate modular conditions independently.

Determine the condition for

n^2 - 1

to be divisible by 8.

n

is even,

n^2 - 1

is odd and cannot be divisible by 8. If

n

is odd, let

n = 2k + 1

; then

n^2 - 1 = 4k(k + 1)

. Since one of

k

k + 1

is always even,

4k(k + 1)

is divisible by 8. Thus,

n

must be odd.

Analyze parity requirements for the power of 2.

Determine the condition for

n^2 - 1

to be divisible by 3.

n

is a multiple of 3,

n^2 - 1 \equiv -1 \pmod 3

, which is not divisible by 3. If

n

is not a multiple of 3,

n \equiv 1

2 \pmod 3

, so

n^2 \equiv 1 \pmod 3

and

n^2 - 1

is divisible by 3. Thus,

n

cannot be a multiple of 3.

Analyze remainder properties modulo 3.

Combine the conditions and count valid positive integers

n < 30

The combined requirement is

\gcd(n, 6) = 1

(

n

is odd and not a multiple of 3). The positive integers less than 30 satisfying this are 1, 5, 7, 11, 13, 17, 19, 23, 25, and 29. Counting these yields 10 integers.

Enumerate all integers meeting the coprime condition within the specified domain.

Anahtar Kavram

Modular arithmetic properties of quadratic expressions and coprimality constraints

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