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
.