Before going to learn the Proof, We know the following
☆Euclid Division Algorithm
For any pair of positive integers a and b (a＞b), there exist a unique pair q and r such that a=bq+r, 0≤ r ＜b
☆ (a + b)2 = a2 + 2ab + b2
☆ (a - b)2 = a2 - 2ab + b2
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☆Show that the Square of a positive integer is in the form of 5p, 5p+1, 5p+4.
Proof:
By Euclid Division Algorithm, we have
Pair of positive integers a and b (a＞b), there exist a unique pair q and r such that a=bq+r, 0≤ r ＜b
Let "a" be the positive integer and take b=5
∴ a=5q+r, 0≤ r ＜5
a=5q+r, (values of r may be 0 or 1 or 2 or 3 or 4)
Do Square on both sides
(a)2=(5q+r)2 [apply (a + b)2 = a2 + 2ab + b2 formula in RHS]
a2 = (5q)2 + 2(5q)(r) + (r)2 [Here a = 5q and b = r ]
[ Substitute different "r" values]
Case-i: Take r=0
a2 = (5q)2 + 2(5q)(0) + (0)2
a2 = (5q)2 + 0 + 0
a2 = 52q2
a2 = 25q2
a2 = 5(5q2)
a2 = 5P (Here P= 5q2 )
Case-ii: Take r=1
a2 = (5q)2 + 2(5q)(1) + (1)2
a2 = 52q2 + 10q + 1
a2 = 25q2 + 10q + 1
Factorize first two terms of RHS
a2 = 5 x 5 x q2 + 5 x 2 x q + 1
Take 5 as Common from first two terms
a2 = 5 (5q2 + 2q) + 1
a2 = 5P + 1 (Here P = 5q2 + 2q )
Case-iii: Take r=2
a2 = (5q)2 + 2(5q)(2) + (2)2
a2 = 52q2 + 20q + 4
a2 = 25q2 + 20q + 4
Factorize first two terms of RHS
a2 = 5 x 5 x q2 + 5 x 2 x 2 x q + 4
Take 5 as Common from first two terms
a2 = 5 (5q2 + 4q) + 4
a2 = 5P + 4 (Here P = 5q2 + 4q )
Case-iv: Take r=3
a2 = (5q)2 + 2(5q)(3) + (3)2
a2 = 52q2 + 30q + 9
a2 = 25q2 + 30q + 9
Factorize first two terms of RHS
And write 9 as 5+4
a2 = 5 x 5 x q2 + 5 x 2 x 3 x q + 5 x 1 + 4 [ 5=5 x 1]
Take 5 as Common from first three terms
a2 = 5 (5q2 + 6q + 1) + 4
a2 = 5P + 4 (Here P = 5q2 + 6q + 1)
Case-v: Take r=4
a2 = (5q)2 + 2(5q)(4) + (4)2
a2 = 52q2 + 40q + 16
a2 = 25q2 + 40q + 16
Factorize first two terms of RHS
And write 16 as 15+1
a2 = 5 x 5 x q2 + 5 x 2 x 2 x 2 x q + 5 x 3 + 1 [ 15=5 x 3]
Take 5 as Common from first three terms
a2 = 5 (5q2 + 8q + 3) + 1
a2 = 5P + 1 (Here P = 5q2 + 8q + 3)
From above five cases, we to conclude that
"Square of every positive integer is in the form of 5P or 5P+1 or 5P+4"
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Click the link to see the Proof of "Cube of every positive integer is in the form of 9m or 9m+1 or 9m+8"
Model Problems for Practice:
1. Show that the square of any positive integer is in the form of 3P or 3P+1 using Euclid Division Algorithm.
2. Show that the square of any positive integer is in the form of 4P or 4P+1 using Euclid Division Algorithm.
3. Show that the square of any positive integer is in the form of 7P, 7P+1, 7P+2 or 7P+4 using Euclid Division Algorithm.