6. Show that any positive odd integer is of the form 8q + 1, 8q +3,89

1.	6. Show that any positive odd integer is of the form 8q + 1, 8q +3,89 +5 or89 + 7, where q is some integer.[HOTS]
Answer» Let a be any positive integer and b =8 ,then , ecluid algorithm , a = 8q +r for some integer q >or = 0 and r = 0,1,2,3,4,5,6,7 0 <,= r <= 7 then , 8q +1 = 2 x 4q +1 2m +1, where m is any positive integer = 4q 8q+3 = 8q +2+1 2 (4q +1) +1 => 2 m +1 , where m is any positive integer = 4q +1 8q +5 = 8q + 4+1 2 (4q +2) +1 =2 m +1 , where m is any positive integer = 4q+2 8q +7 = 8q + 6+1 2( 4q +3) +1 =2m +1 , where m is any positive integer = 4q +3 this means that 8q +1, 8q+3 , 8q+5 and 8q+7 cannot be exactly divisible by 2 . Hence any odd no. can be expressed in the form of 8q+1 , 8q+3 , 8q+5 and 8q+7

6. Show that any positive odd integer is of the form 8q + 1, 8q +3,89 +5 or89 + 7, where q is some integer.[HOTS]

Answer»

Let a be any positive integer and b =8 ,then , ecluid algorithm ,

a = 8q +r for some integer q >or = 0 and r = 0,1,2,3,4,5,6,7 0 <,= r <= 7

then , 8q +1 = 2 x 4q +1 2m +1, where m is any positive integer = 4q

8q+3 = 8q +2+1 2 (4q +1) +1 => 2 m +1 , where m is any positive integer = 4q +1

8q +5 = 8q + 4+1 2 (4q +2) +1 =2 m +1 , where m is any positive integer = 4q+2

8q +7 = 8q + 6+1 2( 4q +3) +1 =2m +1 , where m is any positive integer = 4q +3

this means that 8q +1, 8q+3 , 8q+5 and 8q+7 cannot be exactly divisible by 2 .

Hence any odd no. can be expressed in the form of 8q+1 , 8q+3 , 8q+5 and 8q+7