Number System

Let numbers be 15x and 15y where gcd(x,y)=1. Product = 15x * 15y = 225xy = 6300. xy = 6300/225 = 28. Pairs (x,y) such that gcd(x,y)=1 are (1,28) and (4,7). There are 2 pairs.

HCF = 21. Numbers are 21a and 21b where a, b are coprime. LCM = 21ab = 4641. ab = 4641/21 = 221. Factors of 221 are 13 * 17. So numbers are 21*13 = 273 and 21*17 = 357.

To find the greatest number, subtract the remainder 12 from each number: 1344, 1856, and 2752. The greatest number is the HCF of these three values. The HCF of 1344, 1856, and 2752 is 64.

Find GCD of (3026-11) and (5053-13) = GCD(3015, 5040). 3015 = 45 * 67, 5040 = 45 * 112. The GCD is 45.

The LCM of 4, 7, and 13 is 364. The greatest four-digit number is 9999. Dividing 9999 by 364 gives 27 with a remainder of 183. 9999 - 183 = 9816. Adding the remainder 3 gives 9819. Checking options, 9831 is not divisible by 364 with remainder 3. 9831 / 4 = 2457 R 3, 9831 / 7 = 1404 R 3, 9831 / 13 = 756 R 3. The calculation 9831 is correct.

The product of two numbers equals the product of their HCF and LCM. 27 * x = 9 * 459. x = (9 * 459) / 27 = 459 / 3 = 153.

N = LCM(5, 6, 8) * k + 2 = 120k + 2. For 3-digit values, 100 <= 120k + 2 <= 999. 98 <= 120k <= 997. k can be 1, 2, 3, 4, 5, 6, 7, 8. Total 8 values.

Find the differences: 135-63 = 72, 243-135 = 108, 243-63 = 180. The HCF of 72, 108, and 180 is 36.

To find the largest number that divides 70 and 125 with remainders 5 and 8, subtract the remainders first: 70 - 5 = 65 and 125 - 8 = 117. The largest number is the greatest common divisor (GCD) of 65 and 117, which is 13.

For divisibility by 11, the difference between the sums of alternating digits must be a multiple of 11. With total digit sum 31, the required alternating sums would be 21 and 10, but an alternating sum of 21 is impossible for two digits. Therefore, the fraction is 0, which is not listed.

The greatest number dividing a, b, c leaving the same remainder is gcd(|a-b|, |b-c|, |c-a|). |221-116| = 105. |356-221| = 135. |356-116| = 240. gcd(105, 135, 240) = 15.

The number ends in ...5354. A number is divisible by 8 if its last three digits are divisible by 8. The last three digits are 354. 354 / 8 = 44 with a remainder of 2.

If two numbers leave the same remainder when divided by n, their difference must be divisible by n. The difference is 34041 - 32506 = 1535. The factors of 1535 are 1, 5, 307, 1535. Since n is a three-digit integer, n = 307.

Number N = 3(4(7k + 4) + 1) + 2 = 3(28k + 16 + 1) + 2 = 3(28k + 17) + 2 = 84k + 51 + 2 = 84k + 53. Dividing by 84 gives a remainder of 53.

Two-digit numbers with remainder 3 when divided by 7 are of the form 7n + 3. Smallest is 10 (7*1 + 3), largest is 94 (7*13 + 3). This is an arithmetic progression: 10, 17, ..., 94. Number of terms = (94-10)/7 + 1 = 13. Sum = (n/2)(first + last) = (13/2)(10 + 94) = 13/2 * 104 = 13 * 52 = 676.