Question

show that the square of any positive integer can't be of the form 5m+2 or 5m+3 where M is a whole number ​

Answer 1

Answer:

==> a = 5( 5q² + 8q + 3 ) + 1 . °• a = 5m + 1 . [ Where m = 5q² + 8q + 3 ] . → Therefore, square of any positive integer in cannot be of the form 5m + 2 or 5m + 3 .

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Answer 2

Given :    square of any positive integer  

To Find :  show that the square of any positive integer can't be of the form 5m+2 or 5m+3

Solution:

with out losing generality any number can be of form

5k , 5k+1 , 5k+2 , 5k+3 , 5k+4

(5k)² = 25k² = 5(5k²) = 5m

(5k+1)² = 25k² + 10k + 1  = 5(5k² + 2k)  + 1 = 5m+1

(5k+2)² = 25k² + 20k + 4  = 5(5k² + 4k)  + 4 = 5m+4

(5k+3)² = 25k² +30k + 9  =  25k² +30k + 5 + 4  = 5(5k² + 6k + 1 )  + 4 = 5m+4

(5k+4)² = 25k² + 40k + 16  = 25k² +40k + 15 + 1  =  5(5k² + 8k+3)  + 1 = 5m+1

5m , 5m+1  & 5m + 4   are the only

Hence 5m+2 or 5m+3  can't be of the form

QED

square of any positive integer can't be of the form 5m+2 or 5m+3

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