Question

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FIND HCF OF 27727 AND 53124​

Answer 1

Given two numbers are,

27,727 and 53,124

To find,

HCF(27,727 , 53,124) = ?

We'll find the HCF of the given two numbers by using Euclid's division algorithm.

But before we go on, let's understand the rules of using Euclid's division algorithm.

Step 1: Pick out the greatest number from the given any two numbers.

Step 2: Divide the the greater number by the smaller number and write it in the form of Euclid's division lemma, which is,

$\boxed{a\:=\:b \times q + r}$

Where $\boxed{0 \leq r < b}$

Where a is the dividend, b is the divisor, q is the quotient and r is the remainder.

Step 3: If the remainder is not equal to 0, then, divide the divisor with the remainder. They'll be our new dividend and the divisor.

Step 4: If the remainder ≠ 0, then continue the steps as above said in step 3.

Step 5: If the remainder = 0, then the divisor of that step is the HCF of the given two numbers.

Also let's understand the word algorithm and lemma.



Algorithm : -

Algorithm is a step by step process to solve any problem.

Lemma : -



Lemma is the proven statement which is used to prove another statement. 

Now, we've learnt 'bout the steps and now we're abled to solve our question.

Divide 53,124 and 27,727.

By usin' Euclid's division lemma,

53,124 = 27,727 * 1 + 25,397

Since, the r ≠ 0.

Now, we've to divide 27,727 by 25,397.

By using Euclid's division lemma,

27,727 = 25,397 * 1 + 2,330

Since, the r ≠ 0.

∴ We've to divide 25,397 with 2,330.

By using Euclid's division lemma,

25,397 = 2,330 *  1 + 2097

Since, the r ≠ 0.

Now, we have to divide 2,330 with 2097.

By using Euclid's division lemma,

2,330 = 2097 * 1 + 33

Since, the r ≠ 0.

Now, we've to divide 2097 by 33

By using Euclid's division lemma,

2097 = 33 * 63 + 18

Since, the remainder ≠ 0

Now, we've to divide, 33 by 18

By using Euclid's division lemma,

33 = 18 * 1 + 15

Since, the remainder ≠ 0.

Now, we have to divide 18 by 15.

18 = 15 * 1 + 3

Since, the r ≠ 0.

Now, we've to divide 15 by 3.

15 = 3 * 5 + 0

Since, the r = 0 the divisor of the above step will be the HCF of given two numbers.

∴ 3 is the HCF of 27,727 and 53,124.

$\boxed{\huge{HCF\:=\:3}}$

Answer 2

Prime factors of

27727 = 7×17×119×233

and

53124

= 2×4×3×12×19× 228× 233

so the HCF is 233 as it is in common in factors of both numbers