Name: Rational Numbers Between Two Rational Numbers
Uploaded: 2019-07-19
Duration: 9 min 50 s
Description: In this video, we will learn that we can find infinite rational numbers between any two rational numbers. Interested to know more? Just click!

Question

Use Euclid's division algorithm to find H.C.F of $870$ and $225$ .

Accepted Answer

Euclid's Division Lemma: Given positive integers $a$ and $b$ , there exist unique pair of integers $q$ and $r$ satisfying $a = b q + r,$ such that $0 \leq r < b$ .

Here, given integers $870$ and $225$ are such that $870 > 225$ , we apply Euclid's division algorithm to these integers.

$870 = 225 \times 3 + 195$ ............................. $(1)$

Since, remainder $195 \neq 0$ , we apply the Euclid's division algorithm to $130$ and $30$ to get

$225 = 195 \times 1 + 30$ ............................... $(2)$

From equation $(2)$ it is clear that remainder $30 \neq 0$ so again apply Euclid's division algorithm to $195$ and $30$ , to get

$195 = 30 \times 6 + 15$ ................................ $(3)$

From equation $(2)$ it is clear that remainder $15 \neq 0$ so again apply Euclid's division algorithm to $30$ and $15$ , to get

$30 = 15 \times 2 + 0$ ................................ $(4)$

From equation $(4)$ it is clear that the remainder is zero so, the process is completed.

Since the divisor at this stage is $15$

Hence, the HCF of $870$ and $225$ is $15$ .

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