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2. Show that any positive odd integer is of the form 89 + 1, 89 + 3,89 + 5 or 89 + 7 where q is some integer.. |
| 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 <= 7then , 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 +18q +5 = 8q + 4+1 2 (4q +2) +1 =2 m +1 , where m is any positive integer = 4q+28q +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 | |