Advertisements
Advertisements
Question
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
On dividing n by 5, let q be the quotient and r be the remainder.
Then n = 5q + r, where 0 ≤ r < 5
`\implies` n = 5q + r, where r = 0, 1, 2, 3, 4
`\implies` n = 5q or 5q + 1 or 5q + 2 or 5q + 3 or 5q + 4
Case I: If n = 5q,
Then only n is divisible by 5.
Case II: If n = 5q + 1,
Then n + 4 = 5q + 1 + 4 = 5q + 5 = 5(q + 1) which is divisible by 5
So, in this case, only (n + 4) is divisible by 5.
Case III: If n = 5q + 2,
Then n + 8 = 5q + 10 = 5(q + 2) which is divisible by 5
So, in this case, only (n + 8) is divisible by 5.
Case IV: If n = 5q + 3,
Then n + 12 = 5q + 3 + 12 = 5q + 15 = 5(q + 3) which is divisible by 5
So, in this case, only (n + 12) is divisible by 5.
Case V: If n = 5q + 4,
Then n + 16 = 5q + 4 + 16 = 5q + 20 = 5(q + 4) which is divisible by 5
So, in this case, only (n + 16) is divisible by 5.
Hence 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.
APPEARS IN
RELATED QUESTIONS
Show that every positive even integer is of the form 4m and that every positive odd integer is of the form 4m + 1 for some integer m.
Using prime factorization, find the HCF and LCM of 12,15, 21 .
Without actual division show that each of the following rational numbers is a non-terminating repeating decimal
(i) `73/(2^3× 3^3 × 5)`
Show that the following numbers are irrational.
Prove that \[2 - 3\sqrt{5}\] is an irrational number.
Prove that \[\sqrt{5} + \sqrt{3}\] is irrational.
There are 312, 260 and 156 students in class X, XI and XII respectively. Buses are to be hired to take these students to a picnic. Find the maximum number of students who can sit in a bus if each bus takes equal number of students ______.
“The product of two consecutive positive integers is divisible by 2”. Is this statement true or false? Give reasons.
Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m.
For any positive integer n, prove that n3 – n is divisible by 6.
