Question

Let n>1 be an integer. Which of the following sets of numbers necessarily contain a multiple of 3 ? ​a. n 19 ​-1,n 19 ​+1 b. n 19 ​​,n 38 ​-1 c. n 38 ,n 38 ​+1 d. n 38 , n 19 -1

Answer 1

Suppose the numbers N 1 , N 2 , N 3 .. give quotients Q 1 , Q 2 , Q 3 .. and remainders R 1 , R 2 , R 3 ..., respectively, when divided by a common divisor D.

Therefore N 1 = D × Q 1 + R 1 ,

N 2 = D × Q 2 + R 2 ,

N 3 = D × Q 3 + R 3 .. and so on.

Let P be the product of N 1 , N 2 , N 3 ...

Therefore, P = N 1 N 2 N 3 .. = (D × Q 1 + R 1 )(D × Q 2 + R 2 )(D × Q 3 + R 3 )..

= D × K + R 1 R 2 R 3 ... where K is some number ---- (1)

In the above equation, only the product R 1 R 2 R 3 ... is free of D, therefore the remainder when P is divided by D is the remainder when the product R 1 R 2 R 3 ... is divided by D.

Let S be the sum of N 1 , N 2 , N 3 ...

Therefore, S = (N 1 ) + (N 2 ) + (N 3 ) +...

= (D × Q 1 + R 1 ) + (D × Q 2 + R 2 ) + (D × Q 3 + R 3 )..

= D Ã— K + R 1 + R 2 + R 3 ... where K is some number--- (2)

Hence the remainder when S is divided by D is the remainder when R 1 + R 2 + R 3 is divided by D.