The sum of the ages of my five friends is 47. Their ages are positive

Question

The sum of the ages of my five friends is 47. Their ages are positive integers, and any two of them have a common divisor greater than 1. How old is the eldest of them ?

A. 14
B. 15
C. 18
D. 21
E. 27

Kinshook · Answer

Given: 
1. The sum of the ages of my five friends is 47. 
2. Their ages are positive integers, and any two of them have a common divisor greater than 1.

Asked: How old is the eldest of them ?

Let the ages be a, b, c, d & e
a+b+c+d+e = 47

{6,6,10,10,15}
satisfy the conditions

IMO B

nick1816 · Answer

There is a unique solution of this problem.
2, 3, 6, 12 and 24 or 2, 3, 12, 12 and 18 is not correct.

HCF(2,3) is 1

shameekv1989 · Answer

I could get 15 as one of the answers - not sure if its unique.

If we have 6,6,10,10,15 then we can pick any 2 and there will be a common divisor which will be greater than 1

nick1816  :- Let me know if this is the only solution.

nick1816 · Answer

As 47 is prime, there cannot be a shared factor across the ages of all friends, otherwise their sum would be a multiple of that factor. This rules out having 2 or fewer age groups.

We are also limited to ages with multiple prime factors - if one age group had a single prime factor, the remaining ages must share that prime in order to have a common divisor. This means the youngest our friend can be is 6, and no older than 23 (as 47−4×6=23). We also need at least one friend to be 6, as the average age must be 47/5=9.4

Because of this, we are limited to having 3 different age groups - having 4 or more age groups would require each age to have 3 shared prime factors, ruling out 6 as an option

And finally, we need at least one odd number, otherwise their sum could not be 47. This means either 15 or 21.

This leaves us with two possible sets of primes:

2, 3 & 5

2, 3 & 7 (Not possible)

And so we're left with the former, with the oldest being 15, and the remaining friends being 10, 10, 6 & 6.

MV94 · Answer

I think the answer is 6, 6, 10, 10, and 15, which indicates the answer would be B.

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The sum of the ages of my five friends is 47. Their ages are positive

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The sum of the ages of my five friends is 47. Their ages are positive

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