An integer N has exactly three distinct prime factors p, q, and r, such that N = p^a \cdot… — GMAT

An integer $N$ has exactly three distinct prime factors $p$ , $q$ , and $r$ , such that $N = p^a \cdot q^b \cdot r^c$ , where $a$ , $b$ , and $c$ are positive integers with $a < b < c$ . If $N$ has $24$ positive factors and $N^2$ has $105$ positive factors, what is the value of $a + b + c$ ?

Cevap: 6

Cevap

The value of

a + b + c

6

For an integer with prime factorization

p^a q^b r^c

, the number of positive divisors is

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

. For

N^2 = p^{2a} q^{2b} r^{2c}

, the number of positive divisors is

(2a+1)(2b+1)(2c+1)

. Factoring

105

into three odd terms greater than

1

yields

3 \times 5 \times 7

. Matching these terms in increasing order gives

2a+1=3 \implies a=1

2b+1=5 \implies b=2

, and

2c+1=7 \implies c=3

. Checking

(1+1)(2+1)(3+1) = 24

confirms these values. Summing

a+b+c

yields

1+2+3=6

Adım Adım Çözüm

Write the formula for the number of positive factors of

N

and

N^2

The number of positive factors of

N = p^a q^b r^c

(a+1)(b+1)(c+1) = 24

. Since

N^2 = p^{2a} q^{2b} r^{2c}

, the number of positive factors of

N^2

(2a+1)(2b+1)(2c+1) = 105

The number of divisors of a prime-factored integer is found by adding 1 to each exponent in its prime factorization and multiplying the results.

Factor

105

into three odd factors greater than

1

The prime factorization of

105

3 \times 5 \times 7

. The only way to express

105

as a product of three integers greater than

1

3 \times 5 \times 7

Since

a, b, c

are positive integers, each term

2a+1, 2b+1, 2c+1

must be an odd integer greater than

1

Assign the factors using the inequality condition

a < b < c

Since

a < b < c

, it follows that

2a+1 < 2b+1 < 2c+1

. Therefore:

2a+1 = 3 \implies a = 1

;

2b+1 = 5 \implies b = 2

;

2c+1 = 7 \implies c = 3

Matching the ordered values of

2a+1, 2b+1, 2c+1

with the sorted factors

3, 5, 7

uniquely determines

a, b,

and

c

Verify with the factor count for

N

and calculate the sum

(1+1)(2+1)(3+1) = 2 \times 3 \times 4 = 24

, which matches the given condition. The sum

a + b + c = 1 + 2 + 3 = 6

Verification confirms the solution satisfies all constraints.

Anahtar Kavram

Prime Factorization and Divisor Counting Formula

Tahmini Süre:1m 30s

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