Question 1. Find the L.C.M. of the following numbers using common division method:

i) ii) iii)

iv) v) vi)

Answer:

i)

LCM

ii)

LCM

iii)

LMC

iv)

LCM

v)

LCM

vi)

LCM

Question 2: Use the prime factorization method to find the L.C.M. of the following numbers:

i) ii) iii)

iv) v) vi)

Answer:

i)

Hence LCM

ii)

Hence LCM

iii)

Hence LCM

iv)

Hence LCM

v)

Hence LCM

vi)

Hence LCM

Question 3. Find the H.C.F and L.C.M of the following:

i) and

ii) and

iii) and

iv) and

Answer:

i) and

HCF

LCM

ii) and

HCF

LCM

iii) and

HCF

LCM

iv) and

HCF

LCM

Question 4. Find the H.C.F. and L.C.M of the following fractions:

i) ii) iii)

Note: HCF of a given fraction

LCM of a given fraction

Answer:

i)

HCF of numerators HCF of

HCF of denominators HCF of

LCM of numerators LCM of

LCM of denominators LCM of

HCF of a given fraction

LCM of a given fraction

ii)

HCF of numerators HCF of

HCF of denominators HCF of

LCM of numerators LCM of

LCM of denominators LCM of

HCF of a given fraction

LCM of a given fraction

iii)

HCF of numerators HCF of

HCF of denominators HCF of

LCM of numerators LCM of

LCM of denominators LCM of

HCF of a given fraction

LCM of a given fraction

Question 5. Find the smallest number exactly divisible by and

Answer:

The required number is the LCM of and

LCM

Question 6. Find the least number which when divided by and leaves the same remainder in each case.

Answer:

Required number is the LCM of and plus added to it.

LCM

Hence the number is

Question 7. Find the least number which when diminished by , is divisible by each of the numbers and .

Answer:

Required number is the LCM of and plus added to it.

LCM

Hence the number is

Question 8. Find the smallest number which when increased by is divisible by each of the numbers and .

Answer:

Required number is the LCM of and minus added to it.

LCM

Hence the number is

Question 9. Find the greatest number of four digits which is exactly divisible by each one of the numbers and .

Answer:

The required number must be divisible by LCM of and

The LCM

Now the largest four digit number is

On dividing the by , we get and remainder of

Therefore the number is

Question 10. Five bells begin to toll together and toll respectively at intervals of and seconds. After how much time would they toll together again?

Answer:

Required number is the LCM of and

LCM

Hence all the bells will toll together again in seconds or minutes and seconds.

Question 11. Find the least perfect square number which is divisible by and .

Answer:

Required number is the square of LCM of and

LCM

The required perfect square is

Question 12. Find the least number that should be added to so that the sum is divisible by each one of the numbers and .

Answer:

LCM of , and

and remainder of .

Required number so that the sum is divisible by each of the number , and .

The number is

Question 13. Find the least number of five digits which is exactly divisible by each one of the numbers and .

Answer:

LCM of , and

The smallest digit number is

and remainder of .

Required number so that the sum is divisible by each of the number , and .

Question 14. Three boys start cycling around a circular park from the same point at the same time and in the same direction. If these boys, each cycling at a constant speed, complete a round in min, min and min respectively, then after what time would they meet again.

Answer:

The boys complete the circle in and minutes respectively.

LCM of and minutes or hours minutes.

Question 15. The product of two numbers is and their L.C.M. is . Find their H.C.F.

Answer:

Note: The product of two given numbers Product of HCF and LCM

Therefore HCF HCF

Question 16. The product of two numbers is and their H.C.F. is . Find their L.C.M.

Answer:

Note: The product of two given numbers Product of HCF and LCM

Therefore LCM LCM

Question 17. The H.C.F of two numbers is and their L.C.M is . If one number is , find the other.

Answer:

Note: The product of two given numbers Product of HCF and LCM

Therefore Number Number

Question 18. The sum of the H.C.F and L.C.M of two numbers is . If the L.C.M is times the H.C.F. and one of the numbers is , then find the other number.

Answer:

Note: The product of two given numbers Product of HCF and LCM

and

Now

Question 19. The difference between the L.C.M. and H.C.F of two numbers is . If the L.C.M is times the H.C.F and one of the numbers is , find the other.

Answer:

Note: The product of two given numbers Product of HCF and LCM

and

Now

Question 20. The product of two co-prime numbers is . What would be the L.C.M of these numbers?

Answer:

Co-prime are those numbers where HCF is . So if the product of two co-primes is then the numbers are and . Therefore LCM is .