Refer to the attachment ❤️
Answers
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.
Solution:
We know that ..
a = bq + r
Here .. b = 5 and r = 0, 1, 2, 3, 4.
=> Take b = 5 and r = 0
• n = 5q + 0 = 5q
Divisible by 5
• n + 4 = 5q + 4 + 0 = 5q + 4
Not divisible by 5
• n + 8 = 5q + 8
Not divisible by 5
• n + 12 = 5q + 12
Not divisible by 5
• n + 16 = 5q + 16
Not divisible by 5
______________________________
=> Take b = 5 and r = 1
• n = 5q + 1
Not divisible by 5
• n + 4 = 5q + 4 + 1 = 5q + 5
= 5(q + 1)
Divisible by 5
• n + 8 = 5q + 8 + 1
= 5q + 9
Not divisible by 5
• n + 12 = 5q + 12 + 1
= 5q + 13
Not divisible by 5
• n + 16 = 5q + 16 + 1
= 5q + 17
Not divisible by 5
______________________________
=> Take b = 5 and r = 2
• n = 5q + 2
Not divisible by 5
• n + 4 = 5q + 2 + 4
= 5q + 6
Not divisible by 5
• n + 8 = 5q + 2 + 8 = 5q + 10
= 5(q + 2)
Divisible by 5
• n + 12 = 5q + 2 + 12
= 5q + 14
Not divisible by 5
• n + 16 = 5q + 2 + 16
= 5q + 18
Not divisible by 5
____________________________
=> Take b = 5 and r = 3
• n = 5q + 3
Not divisible by 5
• n + 4 = 5q + 3 + 4
= 5q + 7
Not divisible by 5
• n + 8 = 5q + 3 + 8
= 5q + 11
Not divisible by 5
• n + 12 = 5q + 3 + 12 = 5q + 15
= 5(q + 3)
Divisible by 5
• n + 16 = 5q + 3 + 16
= 5q + 19
Not divisible by 5
_____________________________
=> Take b = 5 and r = 4
• n = 5q + 4
Not divisible by 5
• n + 4 = 5q + 4 + 4
= 5q + 8
Not divisible by 5
• n + 8 = 5q + 4 + 8
= 5q + 12
Not divisible by 5
• n + 12 = 5q + 4 + 12
= 5q + 16
Not divisible by 5
• n + 16 = 5q + 4 + 16 = 5q + 20
= 5(q + 4)
Divisible by 5
____________________________
So, only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5.
____ [ HENCE PROVED]
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Refer the attachment.
