Show that one and only one out of n; n + 2 or n + 4 is divisible by 3, where n is any positive integer. - Mathematics

Advertisements
Advertisements
Sum

Show that one and only one out of n; n + 2 or n + 4 is divisible by 3, where n is any positive integer.

Advertisements

Solution

Consider any two positive integers a and b such that a is greater than b, then according to Euclid's division algorithm:

a = bq + r; where q and r are positive integers and 0 ≤  r < b

Let a = n and b = 3, then

a = bq + r ⇒ n = 3q + r; where 0 ≤ r < 3.

r = 0 ⇒ n = 3q + 0 = 3q

r = 1 ⇒ n = 3q + 1  and r = 2 ⇒ n = 3q + 2

If n = 3q; n is divisible by 3

If n = 3q + 1; then n + 2 = 3q + 1 + 2 = 3q + 3; which is divisible by 3

⇒ n + 2 is divisible by 3

If n = 3q + 2; then n + 4 = 3q + 2 + 4

= 3q + 6; which is divisible by 3

⇒ n + 4 is divisible by 3

Hence, if n is any positive integer, then one and only one out of n, n + 2 or n + 4 is divisible by 3.   

Hence the required result.

  Is there an error in this question or solution?

Video TutorialsVIEW ALL [1]

RELATED QUESTIONS

Without actual division show that each of the following rational numbers is a non-terminating repeating decimal. 

(i) `29/343` 


The LCM of two numbers is 1200, show that the HCF of these numbers cannot be 500. Why ?


Find the greatest number of 6 digits exactly divisible by 24, 15 and 36.


Show that the following numbers are irrational.

\[6 + \sqrt{2}\]

Prove that following numbers are irrationals:

\[\frac{3}{2\sqrt{5}}\]

Show that \[5 - 2\sqrt{3}\] is an irrational number.


Prove that for any prime positive integer p, \[\sqrt{p}\]

 is an irrational number.


What is a composite number?


The exponent of 2 in the prime factorisation of 144, is


Find all positive integers, when divided by 3 leaves remainder 2


Prove that the product of two consecutive positive integers is divisible by 2


When the positive integers a, b and c are divided by 13, the respective remainders are 9, 7 and 10. Show that a + b + c is divisible by 13


If d is the Highest Common Factor of 32 and 60, find x and y satisfying d = 32x + 60y


A positive integer, when divided by 88, gives the remainder 61. What will be the remainder when the same number is divided by 11?


Prove that two consecutive positive integers are always co-prime


Euclid’s division lemma states that for positive integers a and b, there exist unique integers q and r such that a = bq + r, where r must satisfy


Using Euclid’s division lemma, if the cube of any positive integer is divided by 9 then the possible remainders are


Prove that n2 – n divisible by 2 for every positive integer n


When the positive integers a, b and c are divided by 13 the respective remainders is 9, 7 and 10. Find the remainder when a b + + 2 3c is divided by 13


If sum of two numbers is 1215 and their HCF is 81, then the possible number of pairs of such numbers are ______.


Share
Notifications



      Forgot password?
Use app×