Question

show that square of any positive integer is of the form
7p and 7p +1 and 7p+4 for some integer p ( explaintion plz it's urgent)​

Answer 1

Answer:

Step-by-step explanation:

p is some integer

show that square of any positive integer is of the form 7p and 7p +1 and 7p+4 for some integer p.

Let b be the divisor and q be the quotient and r = remainder.

»★Let the positive intiger be a

b = 7

≫By Euclid Division lemma:-

a = bq + r

0≤r<b

✰ a = 7q + r

Where, r = 0, 1 , 2 ,3 ,4 ,5 ,6

»» a = 7q + 0

Squaring on both sides

»» a² = 49q²

»» a² = 7(7q²)

»» a² = 7p⠀⠀⠀⠀⠀⠀ [p = 7q²]

»» a = 7q + 1

Squaring on both sides

»» a² = (7q+1)²

»» a² = 49q² + 1 + 14q

»» a² = 7(7q² +2q)+1

»» a² = 7p+1⠀⠀⠀⠀⠀⠀[p = 7q² + 2q]

»» a = 7q + 2

Squaring on both sides

»» a² = (7q +2) ²

»» a² = 49q² + 4 + 28q

»» a² = 7(7q² +4q)+4

»» a² = 7p + 4⠀⠀⠀⠀⠀⠀[p = 7q² + 4q]

So,

The square of any positive integer is of the form 7p and 7p +1 and 7p+4 for some integer p