Least Common Multiple (LCM) &
Highest Common Factor (HCF)
HCF divides the numbers and numbers
divides the LCM.
Product of LCM*HCF
= L.C.M * H.C.F
= The Product of the two numbers.
L.C.M is always multiple of H.C.F.
LCM & HCF of Fractions:
LCM of fractions
= LCM of numerators/HCF of denominators.
HCF of fractions
= HCF of numerators/LCM of denominators.
Type of Question |
Approach |
Find the least number, which is exactly divisible by x, y, z. |
LCM (x, y, z) |
Find the least number, which when divided by x, y, z,
leaves a remainder ‘r’ in each case. |
LCM (x, y, z) + r |
Find the least number, which when divided by x, y, z
leaves remainders a, b, c respectively. |
Observe, if x – a = y – b = z – c = k
(say).
Then LCM (x, y, z) – k
Else use options. |
Find the greatest number, that will exactly divide x, y,z. |
HCF (x, y, z) |
Find the greatest number, that will divide x, y, z
leaving remainders a, b, c respectively. |
HCF (x – a, y – b, z – c) |
Find the greatest number, that will divide x, y, z
leaving the same remainder in each case. |
HCF (x – y, y – z, z – x) |
Example 1:
A red light flashes 3 times per minute and a green light flashes 5 times in two minutes at regular
intervals. If both lights start flashing at the same time, how many times do they flash together in each
hour?
Solution :
LCM of (20, 24) = Duration after which both the light will flash together.
Example 2:
Three wheels can complete 60, 36, 24 revolutions per minute respectively. There is red spot on each
wheel that touches the ground at time zero. After how much time, all these spots will simultaneously
touch the ground again?
Solution :
Find the time taken by the three wheels to complete one revolution and take their LCM.
Example 3:
A is the set of positive integers such that, when divided by 2, 3, 4, 5, 6 leaves the remainders 1, 2, 3, 4,
5 respectively. How many integers between 0 and 100 belong to set A?
Solution :
The least such number is LCM (2, 3, 4, 5, 6) – 1 = 60 – 1 = 59.The next number is 59 + 60 = 119.
So, only one number lies below 100.
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