WORD PROBLEMS: Greatest Common Divisor/ Least Common Multiple
(with answers)
1. Find the greatest possible length of a rope which can be used to measure exactly the lengths
5 m 13 cm, 7 m 83 cm, 10 m 80 cm.
Solution:
GCD(513, 783, 1080) = 27
2. Three pieces of timber 42 m, 49 m and 63 m long have to be divided into planks of the same
length. What is the greatest possible length of each plank?
Solution:
GCD(42, 49, 63) = 7
3. Find the largest number which can divide 1354, 1866 and 2762 leaving the same remainder
10 in each case.
Solution:
let
x = the largest divisor of the given numbers
1354÷ x=a
10
1866÷ x=b
2762÷x=c
10
10
we can write:
x×a10=1354
x×b10=1866
x×c10=2762
now we subtract 10 from both sides in each equation, we get:
x×a=1344
x×b=1856
x×c=2752
x=GCD 1344,1856, 2752=26 = 64
4. Find the least number of square tiles required to pave the ceiling of a room 15 m 17 cm long
and 9 m 2 cm broad.
Solution:
902=2⋅451=2⋅11⋅41
GCD(902, 1517) = 41 = x
1517=37⋅41
let
x = the length of the side of the square tile
the least number of the tiles = 22⋅37 = 814
5. Find the greatest number of five digits which when divided by 12, 15, 18 and 27 leaves a
remainder 3 in each case.
Solution:
LCM(12, 15, 18, 27) = 540
540×186=100 440 - 6-digit number
540×185=99 900
x=99903
6. Three persons, A, B and C run on a round track. A takes 100 seconds, B takes 110 seconds
and C takes 120 seconds to run a round. If they start together when do they meet again ?
Solution: after
x seconds
x = LCM 100,110, 120=2 3×3×52 ×11=6600 seconds = 110 minutes = 1 hour 50 minutes
7. If the fruits are arranged in groups of 3, 4, 6 or 8, no fruit is left behind. Find the number of
fruits.
LCM(3, 4, 6, 8) = 24
8. Find the greatest 4-digit number which is exactly divisible by each one of the numbers 12,
21, 18 and 28.
Solution:
LCM 12, 21, 18,28=22⋅32⋅7=252
252×40=10080 which is a 5-digit number
252×39=9828 which is the greatest 4-digit number divisible by 12, 21, 18 and 28
9. Five bells begin to toll together and toll respectively at intervals of 6, 7, 8, 9 and 12 seconds.
After how much time will they toll together again ?
Solution:
LCM 6, 7,8, 9, 12=23⋅32⋅7=504 seconds = 8 min 24 sec