## Thursday, June 18, 2009

### A Multiple Problem

I am a two-digit number. When I am divided by 8, there is a remainder of 3. When I am divided by 9, there is a remainder of 4. I am less than 80. What number am I?
Jane, Indonesia
One way is to guess and check. Make an intelligent guess and check if both conditions are met. I can guess 27. 27 divided by 8 gives 3 remainder 3. 27 divided by 9 gives no remainder. So is the number 27?
I think it takes time to guess this way. Let's use logical reasoning. The number is 3 more than a multiple 8 and 4 more than a multiple of 9.
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80.
Three more than a multiple of 8: 11, 19, 27, 35, 43, 59, 67, 75 (no need to try 83 and beyond)
Multiple of 9: 9, 18, 27, 36, 45, 54, 63, 72
Four more than multiple of 9: 13, 22, 31, 40, 49, 58, 67, 76
You got the solution, right?
How about a different way? If you know some algebra, the number is 8m+3 or 9n+4 where m and n are whole numbers. 8m+3 = 9n+4 or 8m = 9n + 1 which gives a possible solution of m = 8 and n = 7. Thus, the number which is 8m+3 can be easily found.
What if the last condition that it is less than 80 is not given?
Can you make up a similar interesting problem for the others to solve?