Question

Form two numbers by interchanging digit place of 325. Prove that
the sum of all these three numbers is divisible by 37.​

Answer 1

Answer:

The given number n can be written as a sum of powers of 1000 as follows.

n = (a2 a1 a0) + (a5 a4 a3)*1000 + (a8 a7 a6)*(1000*1000) +….

As 1000 = (1)(mod 37), 1000 as per congruence relation.

For a positive integer n, two numbers a and b are said to be congruent modulo n, if their difference

(a – b) is an integer multiple of n (that is, if there is an integer k such that a – b = kn). This congruence relation is typically considered when a and b are integers, and is denoted

Hence we can write:

n = { (a2a1a0) + (a5a4a3)* (1) + (a8a7a6)* (1)*(1)+…..}(mod 37),

Thus n is divisible by 37 if and if only if the series is divisible by 37.