Let n = 2^a \cdot 3^b \cdot 5^c, where a, b, and c are positive integers. If the integer \… — GMAT

Let $n = 2^a \cdot 3^b \cdot 5^c$ , where $a$ , $b$ , and $c$ are positive integers. If the integer $\frac{n}{6}$ has exactly $32$ positive divisors and the integer $10n$ has exactly $90$ positive divisors, what is the value of $a + b + c$ ?

A
8
9Answer
C
10
D
11
E
12

Answer

The value of

a + b + c

9

Writing the prime factorizations gives

\frac{n}{6} = 2^{a-1} \cdot 3^{b-1} \cdot 5^c

with

a \cdot b \cdot (c+1) = 32

, and

10n = 2^{a+1} \cdot 3^b \cdot 5^{c+1}

with

(a+2)(b+1)(c+2) = 90

. Solving for positive integers

a

b

, and

c

yields

a = 4

b = 2

, and

c = 3

. Thus, the sum

a + b + c = 9

Step-by-Step Solution

Express the prime factorization of

\frac{n}{6}

and write its total divisor count equation.

\frac{n}{6} = \frac{2^a \cdot 3^b \cdot 5^c}{2 \cdot 3} = 2^{a-1} \cdot 3^{b-1} \cdot 5^c

. The number of positive divisors is

a \cdot b \cdot (c + 1) = 32

Dividing

n

6 = 2 \cdot 3

decreases the exponents of

2

and

3

1

each. The formula for the total number of divisors of a number

p_1^{x} p_2^{y} p_3^{z}

(x+1)(y+1)(z+1)

Express the prime factorization of

10n

and write its total divisor count equation.

10n = (2 \cdot 5) \cdot (2^a \cdot 3^b \cdot 5^c) = 2^{a+1} \cdot 3^b \cdot 5^{c+1}

. The number of positive divisors is

(a + 2) \cdot (b + 1) \cdot (c + 2) = 90

Multiplying by

10 = 2 \cdot 5

increases the exponents of

2

and

5

1

each.

Solve the system of equations for the positive integer exponents

a

b

, and

c

From

a \cdot b \cdot (c + 1) = 32

, test integer factors of

32

. Setting

c + 1 = 4 \implies c = 3

, we get

a \cdot b = 8

. Substituting

c = 3

into

(a + 2)(b + 1)(c + 2) = 90

yields

(a + 2)(b + 1)(5) = 90 \implies (a + 2)(b + 1) = 18

. Testing pairs where

a \cdot b = 8

: if

a = 4

and

b = 2

, then

(4 + 2)(2 + 1) = 6 \cdot 3 = 18

, which satisfies both equations.

Because

a, b, c

are positive integers, factor analysis uniquely pinpoints

a = 4

b = 2

, and

c = 3

Calculate

a + b + c

a + b + c = 4 + 2 + 3 = 9

Summing the derived values of the positive integer exponents.

Key Concept

Prime Factorization and Total Number of Divisors

Estimated Time:2m 0s

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