Show that any positive odd integer is of the form 6 q + 1 , or 6 q + 3 , or 6 q + 5 , where q is some integer.
Solution Using Euclid division algorithm, we know that a=bq+r, 0≤r≤b —-(1) Let a be any positive integer and b=6. Then, by Euclid’s algorithm, a=6q+r for some integer q≥0, and r=0,1,2,3,4,5 ,or 0≤r<6. Therefore, a=6qor6q+1or6q+2or6q+3or6q+4or6q+5 6q+0:6 is divisible by 2, so it is an even number. 6q+1:6 is divisible by 2, but 1 is not divisible by 2 so it is an odd number. 6q+2:6 is divisible by 2, and 2 is divisible by 2 so it is an even number. 6q+3:6 is divisible […]
Show that any positive odd integer of the form 6q+ 1 or 6Q + 3 or 6Q + 5 were q in some integer
Solution Let a be any positive odd integer and b = 6. We can apply Euclid’s division algorithm on a and b = 6. ⇒a=6q+r We know that value of b = 6. ⇒0≤r<6 ∴ All possible values of a are: a = 6q a = 6q+1 a = 6q+2 a = 6q+3 a = 6q+4 […]
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