2005 PMWC Problems/Problem I6

Problem

A group of $100$ people consists of men, women and children (at least one of each). Exactly $200$ apples are distributed in such a way that each man gets $6$ apples, each woman gets $4$ apples and each child gets $1$ apple. In how many possible ways can this be done?

Solution

\[m + w + c = 100\] \[6m + 4w + c = 200\]

Subtracting the second equation from the first, we get $5a + 3b = 100$. Looking at this equation $\mod{5}$, we see that $3b$ must be a multiple of 5, so $5|b$. Thus the choices for $b$ are $5,10,15,20,25,30$, which gives us $6$ possible choices.

See also

2005 PMWC (Problems)
Preceded by
Problem I5
Followed by
Problem I7
I: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
T: 1 2 3 4 5 6 7 8 9 10
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