Question

HCF of two consecutive odd numbers is ____.

Answer 1

Hint: Suppose we have two consecutive odd numbers k and k+2. Since, k and k+2 are odd, there are no common factors between the two. One can also understand it by trying to find H.C.F by repeated division method and get the HCF as 1.

Complete step-by-step answer:

To find the HCF of two consecutive numbers, k and k+2 we will use the method of repeated division.

To find the H.C.F of two numbers (suppose a and b) by division method we follow these steps:

Let a>b. Then we divide a by b. If the remainder is 0 then the H.C.F is b.

If the remainder is not zero, then divide b by the remainder. If the new remainder is 0 then the remainder obtained in the previous step is the H.C.F.

If again, the new remainder is not 0, then the previous divisor becomes the dividend and the new remainder becomes the divisor.

We repeat the steps above until we get a remainder (which will be our new divisor) which on dividing the previous divisor (which is the new dividend) leaves no remainder.

Therefore, using the above steps for k and k+2:

First we divide k+2 by k. We can easily see that

$k+2=k\times 1+2$

Therefore, quotient is 1 and remainder is 2

Now, we divide the divisor k by the remainder 2. But, k is an odd number and by dividing an odd number by 2 we will always get 1 as remainder. i.e.

$k=2n+1$ , where n is a positive integer.

Therefore, the HCF is 1.

Note: To verify our result in the solution, let us take 9 and 11 as two consecutive odd numbers. Now following the steps discussed above, we have:

$\begin{align}

  & 11=9\times 1+2 \\

 & 9=2\times 4+1 \\

 & 2=1\times 2+0

\end{align}$

Therefore, 1 is the H.C.F.

Hence, the result is verified.