Number System

To find the greatest number that leaves the same remainder, calculate the differences between the numbers: 199-166=33, 298-199=99, 298-166=132. The greatest common divisor of 33, 99, and 132 is 33.

Find GCD of (132-62), (232-132), (232-62) = GCD(70, 100, 170) = 10.

HCF * LCM = Product of two numbers. 4 * 576 = 64 * x. x = (4 * 576) / 64 = 576 / 16 = 36.

N = 60k + 49. To find the remainder when divided by 15, we write N = 15(4k) + 45 + 4. Thus, N = 15(4k + 3) + 4. The remainder is 4.

The 2-digit numbers giving a remainder of 3 when divided by 7 are of the form 7n + 3. The smallest such number is 10 (n=1) and the largest is 94 (n=13). This forms an arithmetic progression: 10, 17, ..., 94, with 13 terms. The sum is (n/2) * (first + last) = (13/2) * (10 + 94) = (13/2) * 104 = 13 * 52 = 676.

The number must divide (385-11)=374, (435-10)=425, and (541-14)=527. GCD(374, 425, 527). 374 = 17*22, 425 = 17*25, 527 = 17*31. The GCD is 17.

Number = 195k + 47. We want the remainder when divided by 15. 195 is divisible by 15 (195 = 15 * 13). So the remainder is the same as 47 divided by 15. 47 = 15 * 3 + 2. Remainder is 2.

Subtract the remainder 185 from each number: 1120, 4480, 6720. The greatest number is the HCF of these values. HCF(1120, 4480, 6720) = 1120.

If a number N = 779 * q + 47, we want N mod 19. Since 779 = 19 * 41, 779 is divisible by 19. Thus, N mod 19 = 47 mod 19. 47 = 19 * 2 + 9. The remainder is 9.

Divisors: 20, 25, 35, 40. Remainders: 12, 17, 27, 32. Differences: 20-12=8, 25-17=8, 35-27=8, 40-32=8. The number is LCM(20, 25, 35, 40) - 8. LCM(20, 25, 35, 40) = 1400. 1400 - 8 = 1392.

Let the number be N = 119k + 19. We want the remainder when N is divided by 17. Since 119 = 17 * 7, 119k is divisible by 17. The remainder is the same as 19 divided by 17, which is 2.

The LCM of 9, 12, and 18 is 36. Dividing 1000 by 36 gives 1000 = 36 * 27 + 28. The remainder is 28, so subtracting 28 from 1000 makes it exactly divisible by 36.

Smallest three-digit prime number is 101. 101 divided by 13: 13 * 7 = 91, 101 - 91 = 10. The remainder is 10.