A man and his grandson have the same birthday. For six consecutive years, the man's age was a exact multiple of the boy's age. How old were they at the last birthday?

Let’s call the boy’s age X and the grandfather’s age Y in the first year. Let’s look at Y for a moment. We have to find at least 5 consecutive integers, none of which are prime. Why 5 and not 6? Well, we might find 6, but if we only found 5, then X=1 and the Y = the prime number just before the set of 5 that we found.

Some groups I found in a reasonable grandfatherly age were:

A: (47), 48-52
B: (53), 54-58
C: (61), 62-66
D: (73), 74-78
E: (83), 84-88
F: 90-96.

At first I thought I would have to find 6 non-primes in a row, so I already checked group F and quickly saw by their prime factorization that no matter what the boy’s age was, this group could not be a solution. Most of the prime factorizations were larger primes (like 13-47) times a small prime like 2-5.

Well, in groups A-E the first grandfather age is a prime, so the boys ages over the years must be 1, 2, 3, 4, 5, 6. The first two years in all the grandfather age groups satisfy being divisible by the boy’s age (1 and 2). Another easy year to check is the fifth year, where the grandfather’s age must be divisible by 5. A year is only divisible by 5 if it ends in 0 or 5. The only group where the grandfather’s age ends in a 5 in the fifth year is group C.

I checked the other numbers to confirm it satisfies the solution, and it does. So the answer to the final question is the boy was 6 and the grandfather was 66 on their last birthday that the man’s age was an exact multiple of the boy’s age.

Posted by nikki on 2004-08-31 09:30:39