1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
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

Open in App
Solution

## Method 1: Any positive integer will be of form 5q, 5q + 1, 5q + 2, 5q + 3, or 5q + 4. Case I: If n = 5q n is divisible by 5 Now, n = 5q ⇒ n + 4 = 5q + 4 The number (n + 4) will leave remainder 4 when divided by 5. Again, n = 5q ⇒ n + 8 = 5q + 8 = 5(q + 1) + 3 The number (n + 8) will leave remainder 3 when divided by 5. Again, n = 5q ⇒ n + 12 = 5q + 12 = 5(q + 2) + 2 The number (n + 12) will leave remainder 2 when divided by 5. Again, n = 5q ⇒ n + 16 = 5q + 16 = 5(q + 3) + 1 The number (n + 16) will leave remainder 1 when divided by 5. Case II: When n = 5q + 1 The number n will leave remainder 1 when divided by 5. Now, n = 5q + 1 ⇒ n + 2 = 5q + 3 The number (n + 2) will leave remainder 3 when divided by 5. Again, n = 5q + 1 ⇒ n + 4 = 5q + 5 = 5(q + 1) The number (n + 4) will be divisible by 5. Again, n = 5q + 1 ⇒ n + 8 = 5q + 9 = 5(q + 1) + 4 The number (n + 8) will leave remainder 4 when divided by 5 Again, n = 5q + 1 ⇒ n + 12 = 5q + 13 = 5(q + 2) + 3 The number (n + 12) will leave remainder 3 when divided by 5. Again, n = 5q + 1 ⇒ n + 16 = 5q + 17 = 5(q + 3) + 2 The number (n + 16) will leave remainder 2 when divided by 5. Similarly, we can check the result for 5q+ 2, 5q + 3 and 5q + 4. In each case only one out of n, n + 2, n + 4, n + 8, n + 16 will be divisible by 5. Method 2 : Any positive integer is of the form 5q , 5q + 1 , 5q + 2,5q+3 and 5q+4 here , b = 5 r = 0 , 1 , 2 , 3 , 4 when r = 0 , n = 5q n = 5q ----> divisible by 5 ===> [1] n + 4 = 5q + 4 [ not divisible by 5 ] n + 8 = 5q + 8 [ not divisible by 5 ] n + 6 = 5q + 6 [ not divisible by 5 ] n + 12 = 5q + 12 [ not divisible by 5 ] ------------------------------------------- when r = 1 , n = 5q + 1 n = 5q + 1 [ not divisible by 5 ] n + 4 = 5q + 5 = 5 [q+ 1] ----> divisible by 5 ===> [2] n + 8 = 5q + 9 [ not divisible by 5 ] n + 6 = 5q + 7 [ not divisible by 5 ] n + 12 = 5q + 13 [ not divisible by 5 ] ---------------------------------------------- when r = 2 , n = 5q + 2 n = 5q + 2 [ not divisible by 5 ] n + 4 = 5q + 6 [ not divisible by 5 ] n + 8 = 5q +10 = 5 [q + 2 ] ---> divisible by 5 ====> [3] n + 6 = 5q +8 [ not divisible by 5 ] n + 12 = 5q + 14 [ not divisible by 5 ] ---------------------------------------- when r = 3 , n = 5q + 3 n = 5q + 3 [ not divisible by 5 ] n + 4 = 5q + 7 [ not divisible by 5 ] n + 8 = 5q + 11 [ not divisible by 5 ] n + 6 = 5q + 9 [ not divisible by 5 ] n + 12 = 5q + 15 = 5 [ q + 3 ] ---> divisible by 5 ====> [4] ---------------------------------------------------- when r = 4 , n = 5q + 4 n = 5q + 4 [ not divisible by 5 ] n + 4 = 5q + 8 [ not divisible by 5 ] n + 8 = 5q + 12 [ not divisible by 5 ] n + 6 = 5q + 10 = 5 [ q + 2 ] ---> divisible by 5 ====> [5] n + 12 = 5q + 16 [ not divisible by 5 ] from 1 , 2 , 3 , 4 , 5 its clear that one and only one out of n, n+4, n+8, n+12 and n+6 is divisible by 5

Suggest Corrections
0
Join BYJU'S Learning Program
Related Videos
Euclid's Division Algorithm_Tackle
MATHEMATICS
Watch in App
Join BYJU'S Learning Program