Number System

13601 / 87 = 156.33. 87 * 156 = 13572. 13601 - 13572 = 29.

The number N must divide the difference of the two numbers (2272 - 875 = 1397) to leave the same remainder. The factors of 1397 are 11 and 127. Since N is a 3-digit number, N = 127. The sum of the digits is 1 + 2 + 7 = 10.

The difference between divisor and remainder is constant: 15-5=10, 25-15=10, 35-25=10. The number is LCM(15, 25, 35) - 10. LCM(15, 25, 35) = 525. 525 - 10 = 515.

The number must divide (134-2) and (167-2) exactly. GCD(132, 165). 132 = 33 * 4, 165 = 33 * 5. The greatest common divisor is 33.

All prime numbers greater than 3 are of the form 6k+1 or 6k-1 (which is 6k+5).

The greatest common divisor (GCD) of 18, 24, and 42 is 6. Since 6 divides all three and is the largest such integer, the statement is true.

The number must divide (1025-5)=1020, (1299-7)=1292, and (1575-11)=1564. The greatest such number is the HCF of 1020, 1292, and 1564. HCF(1020, 1292) = 68, and 1564 is divisible by 68.

The first 100 natural numbers include 1. The HCF of any set of numbers including 1 is 1.

The product of two numbers equals the product of their HCF and LCM. Here, 103 * 19261 = 1983883. Checking the options, 1133 * 1751 = 1983883. Both numbers are 4 digits and their HCF is indeed 103.

The number must divide (122-5)=117, (150-7)=143, and (115-11)=104. GCD(117, 143, 104) = GCD(9*13, 11*13, 8*13) = 13.

The product of two numbers equals the product of their HCF and LCM. 30 * x = 120 * 6. x = 720 / 30 = 24.

LCM = (306 * 657) / HCF = (306 * 657) / 9 = 34 * 657 = 22338.

The relationship between two numbers, their HCF, and their LCM is: HCF * LCM = Product of the numbers. Given HCF = 15 and Product = 1650, then 15 * LCM = 1650. Dividing 1650 by 15 gives 110.