A positive integer N has the prime factorization N = 2^a \times 3^b \times 5^c, where a, b… — ACT

A positive integer $N$ has the prime factorization $N = 2^a \times 3^b \times 5^c$ , where $a$ , $b$ , and $c$ are positive integers. The greatest common divisor of $N$ and $360$ has exactly $12$ positive factors, and the least common multiple of $N$ and $360$ has exactly $72$ positive factors. How many positive factors does $N$ have?

Cevap: 36

Cevap

The positive integer

N

has exactly 36 positive factors.

By writing the prime factorization of

360

2^3 \times 3^2 \times 5^1

and expressing the GCD and LCM of

N

and

360

in terms of the minimum and maximum exponents of their prime factors, we establish a system of equations for the exponents. Solving this system yields two cases: either

a = 2

b = 1

, and

c = 5

, or

a = 1

with

(b + 1)(c + 1) = 18

. In both cases, the formula for the number of positive factors of

N

, which is

(a + 1)(b + 1)(c + 1)

, evaluates to exactly 36.

Adım Adım Çözüm

Find the prime factorization of

360

360 = 2^3 \times 3^2 \times 5^1

This allows us to write the greatest common divisor (GCD) and least common multiple (LCM) of

N

and

360

in terms of their prime factors.

Express the number of factors of

\text{GCD}(N, 360)

using the minimum exponents.

(\min(a, 3) + 1)(\min(b, 2) + 1)(\min(c, 1) + 1) = 12

. Since

c \ge 1

, we have

\min(c, 1) = 1

, which simplifies the equation to

(\min(a, 3) + 1)(\min(b, 2) + 1) = 6

The GCD of two numbers is found by taking the minimum of their exponents for each prime factor.

Solve for the possible values of

a

and

b

from the GCD factor equation.

Since

a, b \ge 1

, the only integer pairs for the factors of

6

are: Case 1:

a = 2

and

b = 1

, or Case 2:

a = 1

and

b \ge 2

We analyze the possible values of

\min(a, 3)

and

\min(b, 2)

that multiply to

6

under the constraints

a, b \ge 1

Apply the LCM condition to find the number of factors of

N

for both cases.

For Case 1:

\text{LCM}(N, 360)

has factors count

(3+1)(2+1)(c+1) = 72 \implies c = 5

, giving

d(N) = (2+1)(1+1)(5+1) = 36

. For Case 2:

\text{LCM}(N, 360)

has factors count

(3+1)(b+1)(c+1) = 72 \implies (b+1)(c+1) = 18

, giving

d(N) = (1+1)(b+1)(c+1) = 2 \times 18 = 36

The LCM uses the maximum of the exponents, which allows us to relate the remaining unknown exponents to the total factor count.

Anahtar Kavram

Factors, Multiples, and Prime Factorization

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