Question

One and only one out of n, n+4, n+8, n+12 and n+16 is ……. (where n is any positive integer)
A) Divisible by 5
B) Divisible by 4
C) Divisible by 10
D) Divisible by 12

Answer 1

Hint: In this question since we have to check with the divisibility by 5, 4, 10 and 12 therefore we will be considering n as 5q, 4q, 10q and 12q one by one. Here q can be considered as any positive real number. Then change the given numbers that are n, n + 4, n + 8, n + 12 and n + 16 in terms of q. This will help approaching the problem.

Complete step-by-step answer:
Given numbers are n, n + 4, n + 8, n + 12 and n + 16 where n is any positive integer.
Let n = 5q where q belongs to any positive real number.
So the given numbers become 5q, 5q + 4, 5q + 8, 5q + 12, 5q + 16
Now these numbers is written as
5q, 5q + 4, 5(q + 1) + 3, 5(q + 2) + 2, 5(q + 3) + 1
So as we see that from the above numbers only 5q is exactly divisible by 5 and the rest of the numbers when divided by 5 leaves remainder 4, 3, 2 and 1.
So in the given numbers only n is exactly divisible by 5.
Now let n = 4q where q belongs to any positive real number.
So the given numbers become 4q, 4q + 4, 4q + 8, 4q + 12, 4q + 16
Now these numbers is written as
4q, 4(q + 1), 4(q + 2), 4(q + 3), 4(q + 4)
So as we see that from the above numbers all numbers are exactly divisible by 4.
So in the given numbers n, n + 4, n + 8, n + 12 and n + 16 all are exactly divisible by 4.
Let n = 10q where q belongs to any positive real number.
So the given numbers become 10q, 10q + 4, 10q + 8, 10q + 12, 10q + 16
Now these numbers is written as
10q, 10q + 4, 10q + 8, 10(q + 1) + 2, 10(q + 1) + 6
So as we see that from the above numbers only 10q is exactly divisible by 10 and the rest of the numbers when divided by 10 leaves remainder 4, 8, 2 and 6.
So in the given numbers only n is exactly divisible by 10.
Let n = 12q where q belongs to any positive real number.
So the given numbers become 12q, 12q + 4, 12q + 8, 12q + 12, 12q + 16
Now these numbers is written as
12q, 12q + 4, 12q + 8, 12(q + 1), 12(q + 1) + 4
So as we see that from the above numbers only 12q and 12(q + 1) is exactly divisible by 12 and the rest of the numbers when divided by 12 leaves remainder 4, 8, and 4.
So in the given numbers only n and n + 12 are exactly divisible by 12.
Hence one and only one out of given numbers is exactly divisible by 5 and 10.
Hence option (A) and (C) are the correct answer.

Note:If the numbers would have been given in terms of n that is n, n + 4, n + 8, n + 12 and n + 16 and instead there will be some natural numbers like 5, 7, 15, 20. Then the problem could have been easily approached using the concept that divisibility by 5 is depicted if the ending of the number has a 0 or 5 for example 10, 5, 25, 50. The divisibility by 10 can be taken out using the concept that the last digit of the number to be checked must be 0. A number is divisible by 4 if its last digits are divisible by 4. And a number is divisible by 12 if and only if it is completely divisible by 3 as well as 4.