Question

show that the square of any positive integer is of the form pq +1, pq + 4 for some integer q

Answer 1

Step-by-step explanation:

Let 'a' be a positive integer

b=4

According to Euclids division lemma

a=bq+r

a^2=(bq+r)^2

r=0,1,2,3

from 1

for r=0

a^2=(4q+r)^2

a^2=16q^2

a^2=4(4q^2)

=4q where q=4q^2

for r=1

a^2=(4q+1)^2

a^2=16q^2+1+8q

a^2=4(4q^2+2q)+1

=4q+1 where q=4q^2+2q

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