LCM and HCF 1

The least number of five digits which is exactly divisible by 12, 15 and 18, is:

Least number of 5 digits is 10,000 L.C.M. of 12, 15 and 18 is 180.

On dividing 10000 by 180, the remainder is 100.

∴ Required number

= 10000 + (180 – 100) = 10080.

The greatest number of four digits which is divisible by 15, 25, 40 and 75 is:

Greatest number of 4 digits is 9999. L.C.M. of 15, 25, 40 and 75 is 600.

On dividing 9999 by 600, the remainder is 399.

= (9999 – 399) = 9600.

The least number which should be added to 2497 so that the sum is exactly divisible by 5, 6, 4 and 3 is:

L.C.M. of 5, 6, 4 and 3 = 60.

On dividing 2497 by 60, the remainder is 37.

∴ Number to be added

= (60 – 37) = 23.

Find the maximum number of students among whom 429 mangoes and 715 oranges can be equally distributed.

Required number

= HCF of 429 and 715 = 143

Find the greatest number that will divide 115, 149 and 183 leaving remainders 3, 5, 7 respectively.

= HCF of (115 – 3), (149 –5) and (183 – 7)

= HCF of 112, 144 and 176 = 16

Find the greatest number which when subtracted from 3000 is exactly divisible by 7, 11, 13.

= 3000 – LCM of 7, 11, 13

= 3000 – 1001 = 1999

The L.C.M. of two number is 630 and their H.C.F. is 9. If the sum of numbers is 153, their difference is

Let numbers be x and y.

Product of two numbers = their (LCM × HCF)

⇒ xy = 630 × 9

Also, x + y = 153 (given)

since x – y

⇒

Product of two co-prime numbers is 117. Their L.C.M. should be:

H.C.F of co-prime numbers is 1.

So, L.C.M. = 117/1 = 117.

What is the smallest number which when increased by 5 is completely divisible by 8, 11 and 24?

= LCM of ( 8, 11, 24 ) – 5

= 264 – 5 = 259

If the L.C.M and H.C.F. of two numbers are 2400 and 16, one number is 480; find the second number.

Product of numbers

= (LCM × HCF)

⇒ 480 × second number = 2400 × 16

⇒ second number = 80

The LCM of two numbers is 4800 and their HCF is 160. If one of the numbers is 480, then the other number is:

Product of numbers = HCF × LCM

⇒ The other number

=

The LCM of two numbers is 280 and their ratio is 7: 8. The two numbers are:

Let the numbers be 7x and 8x.

⇒ Their HCF = x

Now, LCM × HCF = Product of Numbers

i.e. 280 × x = 56x²

⇒ x = 5

Hence, the numbers are 35 and 40.

The LCM and HCF of two numbers are 84 and 21, respectively. If the ratio of two numbers be 1: 4, then the larger of the two numbers is:

Let the numbers be x and 4x.

Then, 84 × 21 = x × 4x

⇒ 4x² = 1764

⇒ x² = 441

⇒ x = 21

⇒ 4x = 4 × 21 = 84

Thus the larger number = 84

How many numbers, between 1 and 300 are divisible by 3 and 5 together?

LCM of 3 and 5 = 15

∴ 300/15 = 20 numbers

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