Number System

Find LCM of 18, 21, 30, 84. 18=2*3^2, 21=3*7, 30=2*3*5, 84=2^2*3*7. LCM = 2^2 * 3^2 * 5 * 7 = 4 * 9 * 5 * 7 = 1260. The number is 1260 + 5 = 1265.

Let the number be N = 361k + 47. We want N mod 19. Since 361 = 19 * 19, 361k is divisible by 19. Thus, N mod 19 = 47 mod 19. 47 = 19 * 2 + 9. The remainder is 9.

Find LCM of 15, 20, 36, 48. 15 = 3*5, 20 = 2^2*5, 36 = 2^2*3^2, 48 = 2^4*3. LCM = 2^4 * 3^2 * 5 = 16 * 9 * 5 = 720. Number = LCM + remainder = 720 + 3 = 723.

M is divisible by 20, so M = 20k. (M+4)(M+3) = (20k+4)(20k+3) = 400k^2 + 60k + 80k + 12 = 400k^2 + 140k + 12. Dividing by 20, the remainder is 12.

Find the HCF of 144, 180, and 192. 144 = 12 * 12, 180 = 12 * 15, 192 = 12 * 16. The greatest common divisor is 12.

N = 11a + 9 and N = 9b + 7. This means N + 2 is divisible by both 11 and 9 (LCM = 99). So N + 2 = 99k - 2. For the largest 3-digit number, 99*10 - 2 = 988. 988/11 = 89 rem 9, 988/9 = 109 rem 7. Sum of digits 9+8+8 = 25.

The product of two numbers is equal to the product of their HCF and LCM. 275 * x = 11 * 7700. x = (11 * 7700) / 275 = 84700 / 275 = 308.

The number is calculated using the formula: dividend = (divisor * quotient) + remainder. Substituting the given values, we get (54 * 86) + 13 = 4644 + 13 = 4657.

Find the HCF of (140-4), (176-6), and (264-9), which are 136, 170, and 255. The HCF of 136, 170, and 255 is 17.

Let the numbers be 21a and 21b where a and b are coprime. 21 * a * b = 3003, so a * b = 143. Factors of 143 are (1, 143) and (11, 13). Since numbers > 21, we use (11, 13). Numbers are 21*11=231 and 21*13=273. Sum = 504.

n = 5k + 4. Then 2n = 2(5k + 4) = 10k + 8 = 10k + 5 + 3 = 5(2k + 1) + 3. The remainder when 2n is divided by 5 is 3.

We need the greatest number that divides (391 - 7) = 384 and (318 - 6) = 312. The HCF of 384 and 312 is 24.

Find the LCM of 20, 42, 76. LCM(20, 42, 76) = 7980. The number is LCM + remainder = 7980 + 7 = 7987.

If N = 51k + 35, then N = 17(3k) + 17(2) + 1. Thus, N = 17(3k + 2) + 1. The remainder when divided by 17 is 1.

The differences between divisors and remainders are constant: 35-18=17, 45-28=17, 55-38=17. The number is LCM(35, 45, 55) - 17. LCM(35, 45, 55) = 3465. 3465 - 17 = 3448.