Number System

The numbers are 10, 17, ..., 94. This is an arithmetic progression with a = 10, d = 7, and last term l = 94. 94 = 10 + (n-1)7 => 84 = (n-1)7 => n-1 = 12 => n = 13. Sum = (n/2)(a + l) = (13/2)(10 + 94) = (13/2)(104) = 13 * 52 = 676.

The differences between the numbers are 73-25=48, 97-73=24, 97-25=72. The HCF of 48, 24, and 72 is 24.

The highest power of a prime p dividing n! is given by Legendre's formula: sum of floor(n/p^k). For 100! and p=2, this is floor(100/2) + floor(100/4) + floor(100/8) + floor(100/16) + floor(100/32) + floor(100/64) = 50 + 25 + 12 + 6 + 3 + 1 = 97.

Differences between divisors and remainders: 7-5=2, 10-8=2, 15-13=2, 21-19=2. The constant difference is 2. The number is LCM(7, 10, 15, 21) - 2. LCM(7, 10, 15, 21) = 210. 210 - 2 = 208.

The common difference between the divisors (15, 18, 21, 24) and remainders (11, 14, 17, 20) is 4. We find the LCM of 15, 18, 21, 24, which is 2520. The largest 4-digit number is 9999; 9999 divided by 2520 leaves a remainder of 2439. Subtracting 2439 from 9999 gives 7560, and subtracting the common difference 4 gives 7556.

The number must divide (989 - 5) = 984 and (1327 - 7) = 1320. The HCF of 984 and 1320 is 24.

To find an equivalent fraction to 5/6 with a numerator of 25, multiply both the numerator and the denominator by 5. 5*5 = 25 and 6*5 = 30. The fraction is 25/30.

If a number N = 143k + 31, then N = (11 * 13)k + (2 * 13 + 5). N = 13 * (11k + 2) + 5. The remainder when divided by 13 is 5.

If a number N = 259q + 139, then N = (37 * 7)q + (37 * 3 + 28). N = 37(7q + 3) + 28. The remainder when divided by 37 is 28.

Let the number be 5k + 3. The square of the number is (5k + 3)^2 = 25k^2 + 30k + 9. Dividing this by 5, the remainder is the same as 9 divided by 5, which is 4.

N must be a divisor of the difference 68488 - 67516 = 972. The prime factorization of 972 is 2^2 * 3^5. The divisors of 972 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 243, 324, 486, 972. Three-digit divisors are 108, 162, 243, 324, 486, 972, which are 6 values.

m = 5k + 3. Then 20m = 20(5k + 3) = 100k + 60. Dividing 100k + 60 by 25: 100k is divisible by 25, and 60 divided by 25 leaves a remainder of 10.

Let numbers be 8x and 5x. LCM of 8x and 5x is 40x. Given 40x = 360, so x = 9. The numbers are 8*9=72 and 5*9=45. The larger number is 72.

Find the LCM of 12, 16, and 18, which is 144. To leave a remainder of 5, the number must be 144 + 5 = 149.