KVPY SA Exam 2015 Question Paper

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Question : 13 of 80

Marks: +1, -0

Let n > 1 be an integer. Which of the following sets of numbers necessarily contains a multiple of 3 ?

$n^19-1,n^19+1$
$n^19,n^38-1$
$n^38,n^38+1$
$n^38,n^19-1$

Solution:

	case(I) n = 3k	case(II) n = 2k
(A) $n^19-1,n^19+1$	$(3k)^19- 1,(3k)^19 + 1$ not multiple of 3.
(B) $n^19,n^38-1$	$(3k)^19$ = multiple of 3	$(2k)^38- 1$ = multiple of 3
(C) $n^38,n^38+1$	$(3k)^38$ = multiple of 3	$(2k)^38 + 1$ ≠ multiple of 3
(D) $n^38,n^19-1$	$(3k)^38$ = multiple of 3	$(2k)^19- 1$ ≠ multiple of 3

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