2018 AMC 12B Problem 14

Question

Joey and Chloe and their daughter Zoe all have the same birthday. Joey is

1

year older than Chloe, and Zoe is exactly

1

year old today. Today is the first of the

9

birthdays on which Chloe’s age will be an integral multiple of Zoe’s age. What will be the sum of the two digits of Joey’s age the next time his age is a multiple of Zoe’s age?

7

8

9

10

11

Answer: E

Concepts:factor counting divisibility

Difficulty rating: 1870

Small hint:

Let Chloe be

n

today;

y

years from now her age is a multiple of Zoe’s exactly when

1+y

divides

n-1

Big hint:

So

n-1

must have exactly

9

divisors; find the only two-digit such value

Solution:

Let Chloe be

n

today, so she is

n-1

years older than Zoe. In

y

years Chloe’s age

n+y

is a multiple of Zoe’s age

1+y

exactly when

1+y

divides

n-1.

Having

9

such birthdays means

n-1

has exactly

9

divisors.

A number with exactly

9

divisors has the form

p^2q^2

for distinct primes

p,q,

or

p^8.

Because Joey’s age at the requested future birthday has two digits,

n-1\lt99;

the only possibility is

2^2\cdot3^2=36.

So Chloe is

37

and Joey is

38.

Joey’s age

38+y

is a multiple of

1+y

exactly when

1+y

divides

37.

The next time is

y=36,

making Joey

74,

with digit sum

7+4=11.

Thus, the correct answer is E.

2018 AMC 12B Problem 14

Answer: E

Difficulty rating: 1870

Small hint:

Big hint:

Solution:

Problem 14 in other years