Using Euclid's division lemma for any
natural no. and 4 which of the
forms can a positive integer be
represented as where m is an integer?
a) 4m b)4m+ 1 c) 4m+2 d) All of the above
Answers
Answer:
Hi friend,
1.Let positive integer a = 4m + r , By division algorithm we know here 0 ≤ r < 4 , So
When r = 0
a = 4m
Squaring both side , we get
a2 = ( 4m )2
a2 = 4 ( 4m2 )
a2 = 4 q , where q = 4m2
When r = 1
a = 4m + 1
squaring both side , we get
a2 = ( 4m + 1 )2
a2 = 16m2 + 1 + 8m
a2 = 4 ( 4m2 + 2m ) + 1
a2 = 4q + 1 , where q = 4m2 + 2m
When r = 2
a = 4m + 2
Squaring both hand side , we get
a2 = ( 4m + 2 )2
a2 = 16m2 + 4 + 16m
a2 = 4 ( 4m2 + 4m + 1 )
a2 = 4q , Where q = 4m2 + 4m + 1
When r = 3
a = 4m + 3
Squaring both hand side , we get
a2 = ( 4m + 3 )2
a2 = 16m2 + 9 + 24m
a2 = 16m2 + 24m + 8 + 1
a2 = 4 ( 4m2 + 6m + 2 ) + 1
a2 = 4q + 1 , where q = 4m2 + 6m + 2
Hence
Square of any positive integer is in form of 4q or 4q + 1 , where q is any integer . ( Hence proved )
2.Let positive integer a = 5m+ r , By division algorithm we know here 0 ≤ r < 5 , So
When r = 0
a = 5m
Squaring both side , we get
a2 = ( 5m )2
a2 = 5 ( 5m2 )
a2 = 5 q , where q = 5m2
When r = 1
a = 5m + 1
squaring both side , we get
a2 = ( 5m + 1 )2
a2 = 25m2 + 1 + 10m
a2 = 5 ( 5m2 + 2m ) + 1
a2 = 5q + 1 , where q = 5m2 + 2m
When r = 2
a = 5m + 2
Squaring both hand side , we get
a2 = ( 5m + 2 )2
a2 = 25m2 + 5 + 20m
a2 = 5 ( 5m2 + 4m + 5)
a2 = 5q , Where q = 5m2 + 5m + 1
When r = 3
a = 5m + 3
Squaring both hand side , we get
a2 = ( 5m + 3 )2
a2 = 25m2 + 9 + 30m
a2 = 25m2 + 30m + 10- 1
a2 = 5 ( 5m2 + 6m + 2 ) - 1
a2 = 5q -1 , where q = 5m2 + 6m + 2
Hence
Square of any positive integer is in form of 5q or 5q + 4. , where q is any integer . ( Hence proved )
3.Let a be any odd positive integer with b=4.
Therefore,
a= 4q+1 ( By Euclid's Division Lemma)
When a=4q+1
a2=(4q+1)2
=16q2 +1 + 8q
=8(2q2 +q) +1
=8q+1 where (q=2q2+q)
Hence proved.
4.Let a be a given integer.
On dividing a by 6 , we get q as the quotient and r as the remainder such that
a = 6q + r, r = 0,1,2,3,4,5
when r=0
a = 6q,even no
when r=1
a = 6q + 1, odd no
when r=2
a = 6q + 2, even no
when r = 3
a=6q + 3,odd no
when r=4
a=6q + 4,even no
when r=5,
a= 6q + 5 , odd no
Any positive odd integer is of the form 6q+1,6q+3 or 6q+5...
Hope this helps you...
Please mark it as brainliest answer...☺☺☺
Answer:
D.All of the above
Step-by-step explanation:
Because we know that any positive integer is of the form a=bq+r where r is equal to or greater than 0 but smaller than b.
So we have a positive integer a
b=4 ,q=m and r = 0,1,2,3( why? mentioned above)
so every positive integer is of all of the forms given.
i.e 4m,4m+1,4m+2 and 4m+3.
Hope it Helps..