Number System

2-digit numbers leaving remainder 1 when divided by 3 are of the form 3k + 1. The smallest is 10 (3*3+1) and the largest is 97 (3*32+1). This is an arithmetic progression: 10, 13, ..., 97. Number of terms n = (97-10)/3 + 1 = 29 + 1 = 30. Sum = (n/2) * (first + last) = (30/2) * (10 + 97) = 15 * 107 = 1605.

The 2-digit numbers leaving a remainder of 1 when divided by 5 are 11, 16, 21, ..., 96. This is an arithmetic progression with a = 11, d = 5, and last term l = 96. The number of terms n = ((96 - 11) / 5) + 1 = 18. The sum = (n / 2) * (a + l) = (18 / 2) * (11 + 96) = 9 * 107 = 963.

Three-digit numbers leaving remainder 2 when divided by 5 form an arithmetic progression: 102, 107, ..., 997. The number of terms n is found via 997 = 102 + (n-1)5, giving n = 180. The sum is (n/2)(first + last) = (180/2)(102 + 997) = 90 * 1099 = 98910.

Two-digit numbers leaving remainder 2 when divided by 3 are of the form 3n + 2. The sequence is 11, 14, ..., 98. This is an arithmetic progression with a = 11, l = 98, d = 3. Number of terms n: 98 = 11 + (n-1)3 => 87 = (n-1)3 => 29 = n-1 => n = 30. Sum = (n/2)(a + l) = (30/2)(11 + 98) = 15 * 109 = 1635.

Let x be the smaller and y be the larger. x + y = 32. y = 2x + 5. Substitute: x + (2x + 5) = 32 -> 3x = 27 -> x = 9. Then y = 32 - 9 = 23.

To convert a decimal to a fraction, place the number over 1 followed by as many zeros as there are decimal places. 68.347 has 3 decimal places, so it is 68347/1000.

Let the number be 10x + y. (10x + y) = 6(x + y) + 3 => 4x - 5y = 3. (10y + x) = 4(x + y) + 9 => -3x + 6y = 9 => -x + 2y = 3. Solving: x = 2y - 3. 4(2y - 3) - 5y = 3 => 8y - 12 - 5y = 3 => 3y = 15 => y = 5. x = 2(5) - 3 = 7. Number is 75.

If a number N = 48k + 31, then N = 24(2k) + 24 + 7 = 24(2k+1) + 7. Dividing by 24 leaves a remainder of 7.

Subtract the remainder 5 from each number: 256, 928, 1376. The HCF of these numbers is 32.

The number must be of the form LCM(2, 3, 5) * k + 1. LCM(2, 3, 5) = 30. So the number is 30k + 1. For k=1, the number is 31.

If a number n divides both 9238 and 7091 with the same remainder, then n must divide their difference, 9238 - 7091 = 2147. Factoring 2147 gives 19 * 113. Since 113 is a three-digit number, it is the answer.