Question

use euclids divison lemma to show that any positive integer is of the form 6q+1,or6q+3,or 6q+5 , where q is some integers ! explain

Answer 1

By Euclid's division lemma,
a=bq+r , where 0<r<b
since, b=6
Therefore, a=6q+r , where 0<r<6-----( 1.)
It means, r=0,1,2,3,4,5
If r=1 in (1)
a=6q+1 ,,, Which is odd..
If r=3 in (1)
a=6q+3 ,,, Which is odd..
If r=5 in (1)
a=6q+5 ,,, Which is also odd..


May it helps u....

Answer 2

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let a be any positive integer

then

b= 6

a= bq+r

0≤r<b

0≤r<6

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

case 1.

r=0

a= bq+r

6q+0

6q

case 2.

r=1

a= 6q+1

6q+1

case3.

r=2

a=6q+2

case 4.

r=3

a=6q+3

case 5

r=4

a=6q+4

case 6..

r=5

a=6q+5

hence from above it is proved that any positive integer is of the form 6q, 6q+1,6q+2,6q+3,6q+4 and 6q+5and

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