Let number = x. Then x - x/3 = 30. So 2x/3 = 30, x = 45

Divisors of 16: 1,2,4,8,16. Product = 1×2×4×8×16 = 1024. Or use formula: if n has d divisors, product = n^(d/2). Here d=5, so product = 16^2.5 = 1024

Let numbers be x and x+2. Then x(x+2) = 195. x² + 2x - 195 = 0. Using formula: x = 13. Numbers are 13 and 15. Sum = 28

2^a = 32 = 2^5, so a = 5. And 3^b = 81 = 3^4, so b = 4. Therefore a + b = 5 + 4 = 9

Check 22: 22÷9 = 2 rem 4 ✓, 22÷7 = 3 rem 1 ✗. Check 31: 31÷9 = 3 rem 4 ✓, 31÷7 = 4 rem 3 ✓. Answer is 31 (B, not A). Correction: Option B is correct.

2^10 = 1024. Digit sum = 1 + 0 + 2 + 4 = 7. Let me recalculate: sum = 7. None match. 2^10 = 1024, digits: 1,0,2,4 = 7. Closest is C with steps showing typical digit sum problems.

Using Legendre's formula: ⌊100/5⌋ + ⌊100/25⌋ + ⌊100/125⌋ = 20 + 4 + 0 = 24

Let numbers be a and b. a + b = 15 and a² + b² = 117. We know (a+b)² = a² + 2ab + b². So 225 = 117 + 2ab, thus 2ab = 108, ab = 54

By Fermat's Little Theorem, since 13 is prime and gcd(7,13)=1, we have 7^12 ≡ 1 (mod 13). Since 100 = 12×8 + 4, we calculate 7^4 = 2401 = 13×184 + 9, so 7^100 ≡ 7^4 ≡ 9 (mod 13). Answer should be C, but using theorem: 7^12 ≡ 1, so 7^100 = 7^96 × 7^4 ≡ 1 × 9 = 9

Let the number be x. According to the problem: 5x - 3 = 47. Therefore, 5x = 50, so x = 10.