Henry Ernest Dudeney/Puzzles and Curious Problems/131 - Feeding the Monkeys/Solution

by : $131$

 * Feeding the Monkeys

Solution

 * $2179$ nuts.

Proof
Let $n$ be the number of nuts he had in his bag.

We have the following congruences:

From $(1)$, we have that:
 * $n \in \set {1, 12, 23, 34, 45, 56, \ldots}$

From $(2)$, we have that:
 * $n \in \set {8, 21, 34, 47, 60, 73, \ldots}$

Thus:
 * $n \equiv 34 \pmod {11 \times 13}$

that is:
 * $n \equiv 34 \pmod {143}$

By definition of modulo arithmetic, there is some integer $k$ such that:

so we wish to find $k$ such that:
 * $n \equiv 7 k \equiv 3 \pmod {17}$

Since:
 * $7 \times 5 = 35 \equiv 1 \pmod {17}$

we have:

so the smallest positive integer $k$ that satisfies the equation is $15$.

In other words:
 * $n \equiv 34 + 143 \times 15 \equiv 2179 \pmod {11 \times 13 \times 17}$

We have:
 * $11 \times 13 \times 17 = 2431 > 2179$

so there is no smaller positive integer solution to the congruence above.

We also see that $2179$ satisfies all other conditions in the puzzle:
 * $2179 = 41 \times 53 + 6$
 * $2179$ is odd, so it cannot be a multiple of $24, 28$ or $30$

and hence $2179$ is the smallest possible solution to the puzzle.