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Show that the square of any positive int...

Show that the square of any positive integer cannot be of the form `6m+2` or `6m+5` for some integer `q`.

Text Solution

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By Euclid's division algorithm
`a=6q+r`
`"where" 0 le r lt 6`
`"When " , r=0`
`a=6q`
`rArr " "a^(2) =(6q)^(2) =36q^(2) =6 (6q^(2)) =6m " ""(not of the form 6m+2 or 6m +5)"`
Where m =`6q^(2)` is some integer.
`" where " r=1`
`a=6q+1`
`rArr " " a^(2) =(6q+1)^(2)= 36q^(2) +12q+1`
`=6(6q^(2) +2q) +1 =6m+1 " ""(not of the form 6m+2 or 6m+5)"`
`"where" , m =6q^(2) +2q` is some integer.
`"Where "r=2`
`a =6q +2`
`rArr " "a^(2) =(6q+2)^(2) =36q^(2) +24q+4`
`=6(6q^(2)+ 4q)+ 4= 6m+4 " ""(not of the form 6m+2 or 6m +5)"`
Where `m=6q^(2) +6q` is some integer.
when r=3
a=6q+3
`rArr " "a^(2) =(6q+3)^(2) =36q^(2) +36q+9`
`=6(6q^(2) +6q+1) +3 =6m +3 "(not of the form 6m+ 2 or 6m +5)"`
where `m=6q^(2) +6q +1` is some integer
when r=4
a= 6q+4
`rArr " "a^(2) =(6q+4)^(2)=36q^(2) +48q+16`
`=6(6q^(2) +8q+2)+4 =6m +4`(not of the fomr 6m + 2 or 6m +5)
Where `m=6q^(2) + 8q +2` is some integer.
When r=5
a=6q+5
`rArr " "a^(2) =(6q +5)^(2) =36q^(2) +60q+25`
`=6(6q^(2) +10q+4) +1 =6m +1`(not of the form 6m +2 or 6m +5)
where `m=6q^(2) +10q + 4` is some integer.
`:.` the numbers are of the form `6m ,6m+1 ,6m+3,6m +4`
`rArr` the numbers connot be of the form of 6m +2 and 6m+5.
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