Show that one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5, where n is any positive integer [Hint: Any positive integer can be written in the form 5q, 5q + 1, 5q + 2, 5q + 3, 5q + 4]
Solution:
By dividing n by 5,
Consider q as the quotient and r as the remainder.
Now n = 5q + r, where 0 ≤ r < 5
n = 5q + r, where r = 0, 1, 2, 3, 4
n = 5q or 5q + 1 or 5q + 2 or 5q + 3 or 5q + 4.
Case I:
When n = 5q,
then only n is divisible by 5.
Case II:
When n = 5q + 1, then n + 4 = 5q + 1 + 4 = 5q + 5 = 5(q + 1) which is divisible by 5
Only (n + 4) is divisible by 5.
Case III:
When n = 5q + 2, then
n + 8 = 5q + 10 = 5 (q + 2) which is divisible by 5.
Only (n + 8) is divisible by 5.
Case IV :When n = 5q + 3,
n + 12 = 5q + 3 + 12 = + 15 = 5(q + 3) is divisible by 5
Only (n + 12) is divisible by 5.
Case V :When n = 5q + 4,
n + 16 = 5q + 4 + 16
= 5q + 20
= 5(q + 4) is divisible by 5.
Only (n + 16) is divisible by 5.
Therefore, one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5, where n is any positive integer.
✦ Try This: Show that one and only one out of n, n + 2, n + 4, is divisible by 3, where n is any positive integer
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 1
NCERT Exemplar Class 10 Maths Exercise 1.4 Problem 5
Show that one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5, where n is any positive integer
Summary:
One and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5, where n is any positive integer. Hence proved
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