Saved Bookmarks
| 1. |
Show that any positive odd integer is of the form (6q + 1) or (6q + 3) or (6q + 5), where q is some integer. |
|
Answer» As per Euclid’s lemma, We know that, If a and b are two positive integers, then a = bq + r Where q and r are integers such that 0 ≤ r < b. Let a be any positive odd integer and b = 6. ∴ a = 6q + r Where q and r are integers such that 0 ≤ r < 6. ⇒ r = a − 6q Where r is an integer such that 0 ≤ r < 6. ∴ r is odd interger and 0 ≤ r < 6. (∵ a is odd and 6q is even & odd − even = odd) ∴ r can be either 1, 3 & 5. (Because 1, 3 & 5 are only odd integer in between 1 to 6) Case I :- If r = 1, then a = 6q + 1. Case II :- If r = 3 then a = 6q + 3. Case III :- If r = 5 then a = 6q+5. Hence, Any positive odd integer is of the form (6q + 1) or (6q + 5), where q is some integer. Hence Proved. |
|