A positive integer N has exactly 12 positive divisors. If the greatest common divisor of N… — GRE General Test

A positive integer $N$ has exactly $12$ positive divisors. If the greatest common divisor of $N$ and $35$ is $7$ , and $N$ is a multiple of $6$ , which of the following values could be equal to $N$ ? Indicate all such values.

$84$ Cevap
$126$ Cevap
C
$210$
$294$ Cevap
E
$441$

Cevap

The positive integer

N

could be equal to

84

126

, or

294

The integer

N

must contain the prime factors

2

3

, and

7

, but not

5

. For

N

to have exactly

12

divisors, its prime factorization exponent set

\{a,b,c\}

must satisfy

(a+1)(b+1)(c+1) = 12

, which restricts the exponents to a permutation of

\{1, 1, 2\}

. Evaluating the three permutations yields

84

126

, and

294

Adım Adım Çözüm

Analyze the conditions given for

N

Since

N

is a multiple of

6

2 \mid N

and

3 \mid N

. Since

\gcd(N, 35) = 7

7 \mid N

and

5 \nmid N

. Thus,

N

must have prime factors

2, 3, 7

and no prime factor of

5

Establishing the prime factors of

N

based on divisibility and GCD conditions.

Determine the prime factorization form and divisor count.

Let

N = 2^a \cdot 3^b \cdot 7^c

, where

a \ge 1, b \ge 1, c \ge 1

. The number of divisors is given by

(a+1)(b+1)(c+1) = 12

The total number of positive divisors of a prime-factored integer

\prod p_i^{e_i}

\prod (e_i+1)

Find all valid exponent combinations

(a, b, c)

The factors of

12

into three integers each

\ge 2

are

2 \times 2 \times 3

. Therefore, the set of exponents

\{a, b, c\}

must be a permutation of

\{1, 1, 2\}

Each exponent increment

(e+1)

must be at least

2

since every prime

2, 3, 7

is present.

Calculate the possible numerical values of

N

Case 1:

2^2 \cdot 3^1 \cdot 7^1 = 84

.
Case 2:

2^1 \cdot 3^2 \cdot 7^1 = 126

.
Case 3:

2^1 \cdot 3^1 \cdot 7^2 = 294

Evaluating all three possible permutations of exponents for prime bases

2, 3,

and

7

Anahtar Kavram

Prime Factorization, Divisor Count Formula, and GCD Constraints

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru