1. **Problem statement:** We have three types of bowling balls with weights 10, 12, and 14 pounds. Let the number of each type be $x$, $y$, and $z$ respectively, with $x,y,z \geq 1$. The total weight is 200 pounds, $y$ is a multiple of 3, and $x > z$. We want to find the sum of all possible values of $x$. 2. **Set up the equation:** The total weight equation is: $$10x + 12y + 14z = 200$$ with constraints: - $x, y, z \geq 1$ - $y$ is a multiple of 3, so $y = 3k$ for some integer $k \geq 1$ - $x > z$ 3. **Rewrite the equation using $y=3k$:** $$10x + 12(3k) + 14z = 200 \implies 10x + 36k + 14z = 200$$ 4. **Isolate $x$:** $$10x = 200 - 36k - 14z \implies x = \frac{200 - 36k - 14z}{10}$$ Since $x$ must be a positive integer, the numerator must be divisible by 10 and $x \geq 1$. 5. **Constraints summary:** - $k \geq 1$ - $z \geq 1$ - $x = \frac{200 - 36k - 14z}{10} \geq 1$ - $x > z$ 6. **Find integer solutions:** We try values of $k$ starting from 1 upwards, and for each $k$, try values of $z$ starting from 1 upwards, checking if $x$ is integer, positive, and $x > z$. 7. **Check divisibility and inequalities:** - $200 - 36k - 14z$ must be divisible by 10. - $x > z$ means $\frac{200 - 36k - 14z}{10} > z$. 8. **Trial and error:** - For $k=1$: $200 - 36(1) - 14z = 164 - 14z$ - Check divisibility by 10: $164 - 14z \equiv 0 \pmod{10}$ - $164 \equiv 4 \pmod{10}$, $14z \equiv 4z \pmod{10}$ - So $4 - 4z \equiv 0 \pmod{10} \implies 4z \equiv 4 \pmod{10}$ - Possible $z$ values satisfying $4z \equiv 4 \pmod{10}$ are $z=1,6,11,...$ - Since $z \geq 1$, try $z=1$: numerator $=164 - 14(1) = 150$, $x=150/10=15$, check $x > z$? $15 > 1$ true. - $z=6$: numerator $=164 - 84=80$, $x=8$, $8 > 6$ true. - $z=11$: numerator $=164 - 154=10$, $x=1$, $1 > 11$ false. - So for $k=1$, valid $(x,z)$ pairs: $(15,1)$ and $(8,6)$. - For $k=2$: $200 - 72 - 14z = 128 - 14z$ - $128 \equiv 8 \pmod{10}$, $14z \equiv 4z \pmod{10}$ - $8 - 4z \equiv 0 \pmod{10} \implies 4z \equiv 8 \pmod{10}$ - $z=2,7,12,...$ - $z=2$: numerator $=128 - 28=100$, $x=10$, $10 > 2$ true. - $z=7$: numerator $=128 - 98=30$, $x=3$, $3 > 7$ false. - So for $k=2$, valid $(x,z)$: $(10,2)$. - For $k=3$: $200 - 108 - 14z = 92 - 14z$ - $92 \equiv 2 \pmod{10}$, $14z \equiv 4z \pmod{10}$ - $2 - 4z \equiv 0 \pmod{10} \implies 4z \equiv 2 \pmod{10}$ - $z=3,8,13,...$ - $z=3$: numerator $=92 - 42=50$, $x=5$, $5 > 3$ true. - $z=8$: numerator $=92 - 112=-20$ negative, discard. - So for $k=3$, valid $(x,z)$: $(5,3)$. - For $k=4$: $200 - 144 - 14z = 56 - 14z$ - $56 \equiv 6 \pmod{10}$, $14z \equiv 4z \pmod{10}$ - $6 - 4z \equiv 0 \pmod{10} \implies 4z \equiv 6 \pmod{10}$ - $z=4,9,14,...$ - $z=4$: numerator $=56 - 56=0$, $x=0$ not valid. - $z=9$: numerator $=56 - 126=-70$ negative, discard. - No valid $x$ for $k=4$. - For $k=5$: $200 - 180 - 14z = 20 - 14z$ - $20 \equiv 0 \pmod{10}$, $14z \equiv 4z \pmod{10}$ - $0 - 4z \equiv 0 \pmod{10} \implies 4z \equiv 0 \pmod{10}$ - $z=5,10,15,...$ - $z=5$: numerator $=20 - 70 = -50$ negative, discard. - No valid $x$ for $k=5$. 9. **List all valid $(x,y,z)$:** Recall $y=3k$. - $(x,y,z) = (15,3,1)$ - $(8,3,6)$ - $(10,6,2)$ - $(5,9,3)$ 10. **Sum all possible $x$ values:** $$15 + 8 + 10 + 5 = 38$$ **Final answer:** The sum of all possible values of $x$ is **38**.