<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
A positive integer less than 100 leaves remainder 1 when divided by 6 and remainder 4 when divided by 7. Find the sum of all such integers.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
We need x ≡ 1 (mod 6) and x ≡ 4 (mod 7). The numbers that are 1 mod 6 are 1, 7, 13, 19, 25, ...; the smallest of these that is also 4 mod 7 is 25, since 25 = 6·4+1 and 25 = 7·3+4. Solutions repeat every lcm(6,7) = 42, so the values below 100 are 25 and 67. Their sum is 25 + 67 = 92.
The final answer is \boxed{92}.<|im_end|>
<|im_start|>user
A club has 4 times as many seniors as juniors and twice as many sophomores as juniors. If there are 42 members in total and only these three groups, how many seniors are there?
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
Let the number of juniors be x. Then the number of seniors is 4x and the number of sophomores is 2x. The total is x + 4x + 2x = 7x = 42, so x = 6. The number of seniors is 4x = 24.
The final answer is \boxed{24}.<|im_end|>
<|im_start|>user
A right square pyramid has base edge 10 and lateral edge 13. Find the total area of its four triangular faces.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
The slant height runs from the apex to the midpoint of a base edge, not to a corner. Half the base edge is 5 and the lateral edge is 13, so the slant height is √(13² − 5²) = √(169 − 25) = √144 = 12. Each triangular face has area ½ · 10 · 12 = 60, and there are four faces, so the total area is 4 · 60 = 240.
The final answer is \boxed{240}.<|im_end|><|im_start|>user
{your problem}
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant