Question

Use Euclid division lemma to show that any positive odd integer is of the form 6q+1, or
69 + 3 or 6q +5, where q is some integers.​

Answer 1

By Using EUCLID'S DIVISION LEMMA:-

a= bq+r here, 0 < or = to r<b

All possible remainders= 0,1,2,3,4,5,6

ALL integers are possible :-

a=6q+0 Even

a=6q+1 Odd

a=6q+2 Even

a=6q+3 Odd

a=6q+4 Even

a=6q+5 Odd.

Hence Proved.

Answer 2

Answer:

Step-by-step explanation: By Euclid's Division lemma we know that,

a = bq+r where 0<= r< b

Let 'a' be any positive integer and b=6. Then by EDL we get,

a=6q+r where 0<= r< 6

Since remainder 'r' is always greater than or equal to zero or less than divisor '6', therefore we have 6 possible cases

a=6q+0 OR a=6q+1 OR a=6q+2 OR a=6q+3 OR a=6q+4 OR a=6q+5

When a=6q+0 OR a=6q+2 OR a=6q+4, the value of 'a' is a even positive integer

Thus, for the values of a=6q+3 OR a=6q+5, a is an odd positive integer