Solve LCM – HCF

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Q). Let N be the greatest number that will divide 600, 631 and 724 , leaving the same remainder. What is the value of N ?

(a) 41  (b) 37  (c) 33  (d) 31

Ans: (d) 31

Tricks: If a number leaves the same remainder when dividing several numbers, then that number must divide the differences of those numbers exactly.

Find the differences:

• 631 − 600 = 31


• 724 − 631 = 93

Now find the HCF of 31 and 93.

• 31 = 31 × 1


• 93 = 31 × 3

HCF = 31

Therefore, N = 31

Q. If the ratio of two numbers is 5 : 4 and their HCF = 5, find the LCM.

Options: (a) 124 (b) 128 (c) 90 (d) 100

Concept : If two numbers are in ratio a : b, then numbers can be written as: a × HCF and b × HCF

Also, LCM × HCF = Product of two numbers

Solⁿ

Given ratio = 5 : 4
HCF = 5

Numbers = 5×5 and 4×5 (25 and 20)

Now,

LCM = (25 × 20) ÷ 5
LCM = 500 ÷ 5
LCM = 100

Direct Trick : LCM = HCF × (a × b)

So,

LCM = 5 × (5 × 4)
LCM = 5 × 20
LCM = 100

Ans: (d) 100

Tip

• Ratio numbers × HCF = actual numbers


• Use formula: LCM = HCF × (product of ratio terms)

Q.The ratio of two numbers is 3 : 4 and their LCM is 180. What is the bigger number ?

Options: A. 70  B. 40  C. 50   D. 60

Solⁿ

HCF of numbers = LCM​ / Product of ratio terms

Find HCF             

HCF = 180​ / 3 × 4 = 180​/12 = 15

Find the actual numbers :

     First number: 3 × 15 = 45

     Second number: 4 × 15 = 60

Ans: 60

Q.The ratio of two numbers is 5 : 8 and their LCM is 120. What are the two numbers ?

Options: A. 25, 40           B. 20, 32   C. 15, 24     D. 10, 16

Solⁿ

LCM of the numbers = Product of numbers​ / HCF

But an easier rule for ratio:

LCM ÷ (product of ratio terms) = HCF

Find HCF (Common factor)

HCF = 120​/5×8 = 120/40 ​= 3

Find the numbers using ratio

First number = 5×3 = 15

Second number = 8×3 = 24

Ans: 15 and 24

Q. The product of two numbers is 4,235, and their LCM is 385. Find their HCF.
Solⁿ
We know,
Product of two numbers = LCM × HCF

So,
4,235 = 385 × HCF

Therefore,
HCF = 4,235 ÷ 385 = 11

Ans: HCF = 11

Q. The LCM of two numbers is four times their HCF. The sum of their LCM and HCF is 140. If one of the numbers is 112, find the other number.

A. 28  B. 27  C. 26   D. 25

Solution

Let HCF = $x$

Then: $$LCM=4x$$

Given:

$$LCM+HCF=140$$

$$4x+x=140$$

$$5x=140$$

$$x=28$$

So,

HCF = 28 and LCM = 112

Since one number is already 112 (LCM), the other number is: 28​

Ans: A. 28

Trick:

$$HCF=$$140​/4 + 1 = $$28$$

So the other number = 28.