Let N = 2^a \cdot 3^b \cdot 5^c, where a, b, and c are positive integers. The integer N is… — GRE General Test

Let $N = 2^a \cdot 3^b \cdot 5^c$ , where $a$ , $b$ , and $c$ are positive integers. The integer $N$ is divisible by $24$ , has exactly $36$ positive integer divisors, and $\frac{N}{5}$ is not divisible by $25$ . What is the least possible value of $a + b + c$ ?

A
$6$
$7$ Answer
C
$8$
D
$9$
E
$10$

Answer

The correct answer is

7

. Since

24 = 2^3 \cdot 3^1

divides

N

a \ge 3

and

b \ge 1

. The condition that

\frac{N}{5}

is not divisible by

25

restricts

c

to either

1

2

. For

c=2

, the divisor equation becomes

(a+1)(b+1)(3) = 36

, so

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

. Taking

a+1=4

(

a=3

) and

b+1=3

(

b=2

) satisfies

a \ge 3

and gives a minimal sum of

a+b+c = 3+2+2 = 7

Step-by-Step Solution

Analyze divisibility conditions on exponents

Since

N

is divisible by

24 = 2^3 \cdot 3^1

, we must have

a \ge 3

and

b \ge 1

. Since

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

is not divisible by

25 = 5^2

c-1 < 2 \implies c \le 2

. Because

c

is a positive integer,

c

can be

1

2

Establish structural constraints on prime exponents from the divisibility statements.

Set up the total divisor formula

The total number of positive integer divisors of

N

(a+1)(b+1)(c+1) = 36

Apply the standard formula for counting divisors using prime factorization.

Evaluate candidate values for

c = 1

and

c = 2

c = 1

, then

c+1 = 2

, so

(a+1)(b+1) = 18

. Since

a \ge 3 \implies a+1 \ge 4

, possible integer pairs for

(a+1, b+1)

are

(6, 3)

giving

a=5, b=2 \implies a+b+c = 8

, or

(9, 2)

giving

a=8, b=1 \implies a+b+c = 10

.
If

c = 2

, then

c+1 = 3

, so

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

. With

a+1 \ge 4

, possible pairs for

(a+1, b+1)

are

(4, 3)

giving

a=3, b=2 \implies a+b+c = 3+2+2 = 7

, or

(6, 2)

giving

a=5, b=1 \implies a+b+c = 8

Test allowable values of

c

to find all valid exponent triples

(a, b, c)

and minimize their sum.

Identify the minimum sum

The minimum sum of

a + b + c

7

, achieved when

a = 3

b = 2

, and

c = 2

Compare sums across all valid exponent combinations.

Key Concept

Prime Factorization, Divisibility Rules, and Number of Divisors Formula

Estimated Time:2m 0s

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