Question

1. Use Euclid's division algorithm to find the HCF of:
(1) 135 and 225
(ii) 196 and 38220
(ii) 867 and 255​

Answer 1

Given :

1. 135 and 225
2. 196 and 38220
3. 867 and 255

To Find :

• find HCF of the given terms using Euclid's division algorithm.

Concept :

Euclid's division algorithm :- Let a and b be any two positive integers. Then there exists two unique whole numbers q and r such that,

$\boxed{\begin{array}{c}\sf a = bq+r\\\\\sf 0 \leq r < b \end{array}}$

Note :- Start with larger integer.

Solution :

(i) 135 and 225

$\begin{array}{ccccccc}\sf 225&\sf =&\sf 135&\sf \times\sf &\sf 1&\sf +&\sf 90\\\\\sf 135&\sf =&\sf 90&\sf \times &\sf 1&\sf +&\sf 45\\\\\sf 90&\sf =&\sf \underline{\bf 45}&\sf \times &\sf 2&\sf +&\sf 0\end{array}$

(ii) 196 and 38220

$\begin{array}{ccccccc}\sf 38220 &\sf =&\sf \underline{\bf 196}&\sf \times\sf &\sf 195 &\sf +&\sf 0\end{array}$

(iii) 867 and 225

$\begin{array}{ccccccc}\sf 867&\sf =&\sf 225&\sf \times\sf &\sf 3&\sf +&\sf 102\\\\\sf 255&\sf =&\sf 102&\sf \times &\sf 2&\sf +&\sf 51\\\\\sf 102&\sf =&\sf \underline{\bf 51}&\sf \times &\sf 2&\sf +&\sf 0\end{array}$

Required Answer :

1. HCF of 135 and 255 is 45
2. HCF of 196 and 38220 is 196
3. HCF of 867 and 225 is 51

Answer 2

$\bold{(i) 135 \: and \: 225}$

As you can see, from the question 225 is greater than 135.

Therefore, by Euclid’s division algorithm, we have,

$\mapsto{\textrm{{{\color{navy}{225 = 135 × 1 + 90}}}}}$

Now, remainder 90 ≠ 0, thus again using division lemma for 90, we get,

$\mapsto{\textrm{{{\color{navy} {135= 90 × 1 + 45}}}}}$

Again, 45 ≠ 0, repeating the above step for 45, we get,

$\mapsto{\textrm{{{\color{navy} {90 = 45 × 2 + 0}}}}}$

The remainder is now zero, so our method stops here.

Since, in the last step, the divisor is 45, therefore, HCF (225,135) = HCF (135, 90) = HCF (90, 45) = 45.

$\fbox \orange{Hence, the HCF of 225 and 135 is 45.}$

$\bold{(ii) 196 \: and \: 38220}$

In this given question, 38220 >196, therefore the by applying Euclid’s division algorithm and taking 38220 as divisor, we get,

$\mapsto{\textrm{{{\color{navy} {38220 = 196 × 195 + 0}}}}}$

We have already got the remainder as 0 here. Therefore, HCF(196, 38220) = 196.

$\fbox \orange{Hence, the HCF of 196 and 38220 is 196.}$

$\bold{(iii) 867 \: and \: 255}$

As we know, 867 is greater than 255. Let us apply now Euclid’s division algorithm on 867, to get,

$\mapsto{\textrm{{{\color{navy} {867 = 255 × 3 + 102}}}}}$

Remainder 102 ≠ 0, therefore taking 255 as divisor and applying the division lemma method, we get,

$\mapsto{\textrm{{{\color{navy} {255 = 102 × 2 + 51}}}}}$

Again, 51 ≠ 0. Now 102 is the new divisor, so repeating the same step we get,

$\mapsto{\textrm{{{\color{navy} {102 = 51 × 2 + 0}}}}}$

The remainder is now zero, so our procedure stops here. Since, in the last step, the divisor is 51.

Therefore, HCF (867,255) = HCF(255,102) = HCF(102,51) = 51.

$\fbox \orange{Hence, the HCF of 867 and 255 is 51.}$

Thank you!

@itzshivani