Math, asked by honeytejuhoneyteju, 21 days ago

use euclids division lemma to show that the cube of any positive integer is of the form 7m or 7m+1 or 7m+6​

Answers

Answered by pirbushra358
6

Answer:

According to Euclid's Division Lemma

Let us take a as any positive integer and b=7

Then using Euclid's algorithm,

we get a=7q+r ;here r is remainder and value of q is more than or equal to 0 and r=0,1,2,3,4,5,6 because 0≤r<b and the value of b=7

So possible forms will 7q,7q+1,7q+2,7q+3,7q+4,7q+5,7q+6

to get the cube of these values use the formula

(a+b)

3

=a

3

+3a

2

b+3ab

2

+b

3

In this formula value of a is always 7q

so collect the value we get

(7q+b)

3

=343q

3

+147q

2

b+21qb

2

+b

3

Now divide by 7 , we get quotient = 49q

3

+21q

2

r+3qr

2

and remainder is b

3

so we have to consider the value of b

3

b=0 we get 7m+0=7m

b=1 then 1

3

=1 so we get 7m+1

b=2 then 2

3

=8 divided by 7 we get 1 as remainder so we get 7m+1

b=3 then 3

3

=27 divided by 7 we get 6 as remainder so we get 7m+6

b=4 then 4

3

=64 divided by 7 we get 1 as remainder so we get 7m+1

b=5 then 5

3

=125 divided by 7 we get 6 as remainder so we get 7m+6

b=6 then 6

3

=216 divided by 7 we get 6 as remainder so we get 7m+6

so all values are in the form of 7m,7m+1 and 7m+6

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