Thread: Number Thry: Monkeys! (Chinese Remainder Thm)?

1. Number Thry: Monkeys! (Chinese Remainder Thm)?

Suppose that a group of 17 monkeys decide to store all of their bananas in 11 piles, all of equal size, and each of the piles containing more than 1 banana, with a 12th pile of 6 that are left over. No bananas remain when they decide to divide them into 17 equal groups. What is the smallest possible # of bananas that they can have?

Answer in back of book: 204

2. Hello, Ideasman!

This is can be solved with basic Algebra.
. . It took a while to construct a simple approach.

Also, there are far too many words in the problem.

When a number of bananas is divided into 11 equal piles
. . with more than one banana in each pile, there are 6 bananas left over.
However, the bananas can be divided exactly into 17 equal piles.
What is the smallest possible numbers of bananas?

Answer in back of book: 204
Let N be the number of bananas.

Then we are told: . N .= .11a + 6
. . . . . . . . . and: . N .= .17b . . . . . for integers a and b .(a > 1)

So we have: .11a + 6 .= .17b

. . . . . . . . . . . . . .17b - 6 . . . . . .6(b - 1)
Solve for a: . a .= .--------- .= .b + ---------
. . . . . . . . . . . . . . . 11 . . . . . . . . . 11

Since a is an integer, that fraction must also be an integer.

For a > 1, the least value occurs when b = 12.

Then: . a .= .12 + 6(11)/11 .= .18

Therefore: . N .= .11(18) + 6 .= .204