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.
Answers
Answer:
Step-by-step explanation:
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Any positive integer is of the form 5q , 5q + 1 , 5q + 2
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 ]
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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 ]
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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 ]
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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]
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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
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