Contact Style About

15. Use Euclid division lemma to show that the square of any positive integer cannot be of
the form 5m 2 or 5m – 3 for some i

By Emma

15. Use Euclid division lemma to show that the square of any positive integer cannot be of
the form 5m 2 or 5m – 3 for some integer m. (3)​

About the author
Emma

22) Find the ratio of the first quantity to the second quantity
20 second : 1 minute.
I) 1:3
2) 2:3
3) 3:2

draw flowchart to find the value of sum = x/2+x²/3+x³/4+….+xⁿ/x+1.​

2 thoughts on “15. Use Euclid division lemma to show that the square of any positive integer cannot be of<br />the form 5m 2 or 5m – 3 for some i”

  2. Answer:

    Let a be any positive integer

    By Euclid’s division lemma

    a=bm+r

    a=5m+r (when b=5)

    So, r can be 0,1,2,3,4

    Case 1:

    ∴a=5m (when r=0)

    a

    2

    =25m

    2

    a

    2

    =5(5m

    2

    )=5q

    where q=5m

    2

    Case 2:

    when r=1

    a=5m+1

    a

    2

    =(5m+1)

    2

    =25m

    2

    +10m+1

    a

    2

    =5(5m

    2

    +2m)+1

    =5q+1 where q=5m

    2

    +2m

    Case 3:

    a=5m+2

    a

    2

    =25m

    2

    +20m+4

    a

    2

    =5(5m

    2

    +4m)+4

    5q+4

    where q=5m

    2

    +4m

    Case 4:

    a=5m+3

    a

    2

    =25m

    2

    +30m+9

    =25m

    2

    +30m+5+4

    =5(5m

    2

    +6m+1)+4

    =5q+4

    where q=5m

    2

    +6m+1

    Case 5:

    a=5m+4

    a

    2

    =25m

    2

    +40m+16=25m

    2

    +40m+15+1

    =5(5m

    2

    +8m+3)+1

    =5q+1 where q=5m

    2

    +8m+3

    From these cases, we see that square of any positive no can’t be of the form 5q+2,5q+3.

    Reply

Leave a Comment