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.​

Answer 1

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Step-by-step explanation:n,n+4,n+8,n+12,n+16 be  integers.

where n can take the form 5q, 5q+1 ,5q+2 , 5q + 3 , 5q + 4.

Case I when n=5q

Then n is divisible by 5.

but neither of  5q+1 ,5q+2 , 5q + 3 , 5q + 4 is divisible by 5.

Case II when n=5q+1

Then n is not divisible by 5.

n+4 = 5q+1+4 = 5q+5=5(q +1),

which is divisible by 5.(else not)

Case III when n=5q+2

Then n is not divisible by 5.

 n+8 = 5q+2+8 =5q+10=5(q+2),

which is divisible by 5.(else not)

Case IV when n=5q+3

Then n is not divisible by 5.

 n+12 = 5q+3+12 =5q+15=5(q+3),

which is divisible by 5.(else not)

Case V when n=5q+4

Then n is not divisible by 5.

 n+16 = 5q+4+16 =5q+20=5(q+4),

which is divisible by 5.(else not)

Hence, one of n, n+4,n+8,n +12 and n+16 is divisible by 5.

hop it helps u✌️✌️❤️