|
Note that the number of play blocks is a multiple of the LCM of 16, 15, and 14. The value of this can be found to be (16)(15)(7) = 1680. This number is also divisible by 1, 2, 3, 4, 5, 6, 7, 8, 10, and 12, thus, the three numbers x, y, z are 9, 11, 13.
Thus, 1680k ≡3 when taken mod 9, 11, 13. Since
1680 is congruent to 6 mod 9 and 3 mod 13, and
congruent to 8 mod 11, the number k must be a number
that is congruent to 1 mod 13, 2 mod 3 (because 6 is
a multiple of 3, which is a factor of 9 that can be
divided out) and cause 8 to become 3 when multiplied
under modulo 11.
Looking at the last condition shows that k≡ 10 mod 11 (after a bit of bashing) and is congruent to
1 mod 13 and 2 mod 3 as previously noted. Listing out
the numbers congruent to 10 mod 11 and 1 mod 1
yield the following lists:
10 mod 11: 21, 32, 43, 54, 65, 76, 87, 98, 109, 120,
131...
1 mod 13: 14, 27, 40, 53, 66, 79, 92, 105, 118, 131,
144, 157, 170...
Both lists contain x elements where x is the modulo being taken, thus, there must be a solution in these lists as adding 11(13) to this solution yields the next smallest solution. In this case, 131 is the solution for $k$ and thus the answer is 1680(131). Since 131 is prime, the sum of the prime factors is $2 + 3 + 5 + 7 + 131 = 148. |
