Let K be a positive integer with prime factorization K = 2^a \times 5^b \times 11^c, where… — GMAT

Let $K$ be a positive integer with prime factorization $K = 2^a \times 5^b \times 11^c$ , where $a$ , $b$ , and $c$ are positive integers. If $10K$ has 36 more positive divisors than $K$ , and $11K$ has 12 more positive divisors than $K$ , what is the value of $a + b + c$ ?

Answer: 9

Answer

The value of

a + b + c

is 9.

For

K = 2^a 5^b 11^c

, the number of divisors is given by

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

. Multiplying by 11 increases the exponent of 11 by 1, yielding

(a+1)(b+1)

additional divisors. Since this difference is 12,

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

. Multiplying by 10 increases the exponents of both 2 and 5 by 1, yielding an additional

(c+1)(a+b+3)

divisors. Since this difference is 36,

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

. For positive integers

a

and

b

, the only factor pair of 12 that allows

36 / (a+b+3)

to be an integer is when

a+b = 6

. Consequently,

c+1 = 4

, so

c = 3

. Therefore,

a+b+c = 6+3 = 9

Step-by-Step Solution

Write the divisor count formulas for

K

10K

, and

11K

d(K) = (a+1)(b+1)(c+1)

d(11K) = (a+1)(b+1)(c+2)

, and

d(10K) = (a+2)(b+2)(c+1)

The number of positive divisors of a number

2^x 5^y 11^z

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

Use

d(11K) - d(K) = 12

to solve for

(a+1)(b+1)

(a+1)(b+1)(c+2) - (a+1)(b+1)(c+1) = 12 \implies (a+1)(b+1) = 12

Factoring out

(a+1)(b+1)

simplifies the equation directly.

Use

d(10K) - d(K) = 36

to find the relationship between

a, b,

and

c

(c+1)[(a+2)(b+2) - (a+1)(b+1)] = 36 \implies (c+1)(a+b+3) = 36

Expanding

(a+2)(b+2) - (a+1)(b+1) = ab + 2a + 2b + 4 - (ab + a + b + 1) = a + b + 3

Determine

a+b

and

c

using positive integer constraints.

a+b = 6

and

c = 3

Since

a, b \ge 1

, the pairs for

(a+1, b+1)

yielding 12 are

(2,6)

(3,4)

. If

(a+1,b+1)=(2,6)

a+b=6

, making

a+b+3=9

and

c+1=36/9=4 \implies c=3

. If

(3,4)

a+b=5

, so

a+b+3=8

, but

36/8

is not an integer.

Sum

a

b

, and

c

a + b + c = 6 + 3 = 9

Adding the derived sum

a+b=6

and

c=3

gives 9.

Key Concept

Divisor Count Function from Prime Factorization

Estimated Time:2m 0s

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