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Age problem
By Nebiyu Adane on Thursday, October 18, 2001 - 12:07 am:

Hi Guys
Sally is 1/6 as old as her father. Her dad's age, when divided by 2,3,4,6, and 8 leaves one remaining year, but when divided by 5, there is nothing left. How many months old is Sally?

By Dave Sheridan on Thursday, October 18, 2001 - 01:15 pm:

First, do you understand what the question is telling you? The father is a whole number of years old, since his age divided by 5 leaves no remainder. Possibilities include 5, 10, 15, 20, and so on (in fact he's likely to be at least 20 if he has a daughter!)

The other information is that dividing by 2, 3, 4, 6 and 8 gives remainder 1. What does this mean? Well, if he was 31, his age divided by 2 is 15 with remainder 1. Likewise, divided by 3 it's 10 remainder 1. But divided by 4 it's 7 remainder 3.

How do we find the answer other than just by guessing? Well, the father's age must be one more than a multiple of 2,3,4,6 and 8. What's the smallest multiple of these numbers? This is called the lcm (lowest common multiple) by a lot of people. It's 24 - I can explain how to find that if you don't know anything about lcms.

So her father's age is 1+24*x for some x. The possibilities are 25, 49, 73, 97 and so on. The only one of these divisible by 5 without remainder is 25 (unless he's really old...)

But we haven't answered the question yet. We need to find out Sally's age. She's 25/6. An easy way to work this out is to turn this into months first. There are 12 months in a year so her age in months is 25*12/6, which is 25*2=50.

Sally is 50 months old.

-Dave