Question

Show that the square of any positive integer cannot be of the form 5q+2 or 5q+3 for any integer q

Answer 1

Answer:

Number divisible by 5 can be of the form:-

d = 5m + r,

where 0 ≤ r <5

If d = 5m, d² = 5.q,

where q is some integer and q = 5m²

If d = 5m + 1, d² = 5q + 1

If d = 5m + 2, d² = 5q + 4

If d = 5m + 3, d² = 5q + 4

If d = 5m + 4, d² = 5q + 1

Therefore, the square of any positive integer cannot be in the form of 5q + 2 or 5q + 3 for any integer "q".