by using Euclid division algorithm show that the square of any positive integer is of the form 5m or 5m+1 or 5m+4 where 'm' is some integer
Answers
Let x be any integer
Then,
Either x=5m or x=5m+1 or x=5m+2 or, x=5m+3 or x=5m+4 for integer x. [ Using division algorithm]
If x=5m
On squaring both side and we get,
x
2
=25m
2
=5(5m
2
)=5n where n=5m
2
If x=5m+1
On squaring both side and we get,
x
2
=(5m+1)
2
=25m
2
+1+10m
=5(5m
2
+2m)+1(where5m
2
+2m=n)
=5n+1
If x=5m+2
Then x
2
=(5m+2)
2
=25m
2
+20m+4
=5(5m
2
+4m)+4
=5n+4 [ Taking n=5m
2
+4m]
If x=5m+3
Then x
2
=(5m+3)
2
=25m
2
+30m+9
=5(5m
2
+6m+1)+4
=5n+4 [ Taking n=5m
2
+6m+1]
If x=5m+4
On squaring both side and we get,
x
2
=(5m+4)
2
=25m
2
+16+40m
=5(5m
2
+8m+3)+1(where5m
2
+8m+3=n)
=5n+1
Hence, In each cases x
2
is either of the of the form 5n or 5n+1 for integer n..
Step-by-step explanation:Given:
A positive integer of the form 5m+r.
To Show:
square of any positive integer is of the form 5m or 5m +1 or 5m + 4 where m is a whole number.
Solution:
Any number can be represented by the form 5m+r ,
where r can be 0,1,2,3,4 and m ∈ N
Let Q be a positive integer.
By Euclid's Division Lemma,
Q = 5m + r
Squaring Q,
Q² = (5m + r)² = 25m² + 10mr + r²
Q² = 5 ( 5m² + 2mr) + r²
We can take 5m² + 2mr as a number K.
Then Q² becomes,
Q² = 5K + r².
Since r ∈ { 0, 1, 2, 3, 4}
r² ∈ {0,1,4,9,16}
We also have condition that any number of the form aq + r , r ≤ a , since r is the remainder when the number is divided by a.
Therefore r² < 5
Possible values of r² = { 0 , 1, 4 }
Therefore any positive integer is of the form, 5m , 5m +1 or 5m+4.
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