Number System

Let the number be N = 342k + 47. We want N mod 19. Since 342 = 19 * 18, 342k is divisible by 19. Thus, N mod 19 = 47 mod 19. 47 = 19 * 2 + 9, so the remainder is 9.

The number n satisfies n = 3k-1, n = 4m-1, n = 5p-1. Thus n+1 is a multiple of LCM(3, 4, 5) = 60. The smallest positive integer is 60-1 = 59.

The number must divide (309-3)=306, (814-4)=810, and (2219-5)=2214. Find the HCF of 306, 810, 2214. 306 = 18*17, 810 = 18*45, 2214 = 18*123. HCF is 18.

Divisors are 20, 25, 35, 40. Remainders are 14, 19, 29, 34. The difference between divisor and remainder is constant: 6. N = LCM(20, 25, 35, 40) - 6. LCM = 1400. N = 1400 - 6 = 1394.

a_n = 6^n - 5n. For n=1, a_1 = 6-5=1. 1 mod 25 = 1. For n=2, a_2 = 36-10=26. 26 mod 25 = 1. For n=3, a_3 = 216-15=201. 201 mod 25 = 1. The remainder is always 1.

Number = (Divisor * Quotient) + Remainder = (53 * 8) + 5 = 424 + 5 = 429.

The number must divide 64-12=52 and 72-7=65. The largest such number is the GCD of 52 and 65. 52 = 13 * 4, 65 = 13 * 5. GCD = 13.

The number must divide (245 - 5) = 240 and (1029 - 5) = 1024. GCD(240, 1024) = 16.

LCM of 8, 9, 10 is 360. The largest 4-digit number is 9999. 9999 / 360 = 27 with remainder 279. 9999 - 279 = 9720. Adding the remainder 5 gives 9725.

A number is divisible by 16 if its last four digits are divisible by 16. The number ends in ...1920. 1920 / 16 = 120. Since 1920 is divisible by 16, the remainder is 0.

The number is calculated using the division algorithm formula: Dividend = (Divisor * Quotient) + Remainder. Substituting the values gives (9235 * 888) + 222 = 8200680 + 222 = 8200902.

The divisors are 20, 25, 35, 40 and remainders are 14, 19, 29, 34. Note that the difference between divisor and remainder is constant: 20-14=6, 25-19=6, 35-29=6, 40-34=6. The number is LCM(20, 25, 35, 40) - 6. LCM(20, 25, 35, 40) = 1400. 1400 - 6 = 1394.

The number is LCM(4, 6, 10, 15) + 3. LCM(4, 6, 10, 15) = 60. 60 + 3 = 63.