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Example(3)
- June 2, 2020
- Category: Example(3)
(21) Find the lowest number which when subtracted from 10000, is exactly divisible by 15, 27 and 24.
Soln. L.C.M. of 15, 27 and 24 = 1080.
If we divide 10000 by 1080, we get the remainder = 280
i.e. If 280 is subtracted from 10000, then the no. so obtained is divisible by 15, 27 &
(22) The least number which when divided by 57, 95 and 30 leaves the remainder 44, 82 and 17 respectively is :
Soln. Note that the difference between any divisor and the corresponding remainder is the same, which is 13.
Required number = L.C.M. of 57, 95, 30 – 13
= L.C.M. of 570 – 13
= 557
(23) The least number which when divided by 60, 75 and 100 leaves the remainder 52, 67 and 92 respectively is :
Soln. Note that the difference between any divisor and the corresponding remainder is the same, which is 8.
Required number = L.C.M. of 60, 75, 100 – 8
= L.C.M. of 300 – 8
= 292
(24) Find the largest number of four digits exactly divisible by 24, 18, 36 and 30.
Soln. The largest number of four digits is 9999.
Required number must be divisible by L.C.M. of 24, 18, 36 and 30 i.e., 360.
On dividing 9999 by 360, we get 279 as remainder.
Required number = (9999 – 279) = 9720.
(25) Find the largest number of five digits exactly divisible by 20, 16, 25 and 24.
Soln. The largest number of four digits is 99999.
Required number must be divisible by L.C.M. of 24, 18, 36 and 30 i.e., 1200.
On dividing 99999 by 1200, we get 399 as remainder.
Required number = (99999 – 399) = 99600.
(26) Find the smallest number of five digits exactly divisible by 8, 6, 18 and 27.
Soln. Smallest number of five digits is 10000.
Required number must be divisible by L.C.M. of 8, 6, 18, 27 i.e., 216.
On dividing 10000 by 216, we get 64 as remainder.
Required number = 10000 + (216 – 64) = 10152.
(27) Find the smallest number of four digits exactly divisible by 3, 4, 5 and 6.
Soln. Smallest number of five digits is 1000.
Required number must be divisible by L.C.M. of 3, 4, 5, 6 i.e., 60.
On dividing 1000 by 60, we get 40 as remainder.
Required number = 1000 + (60 – 40) = 1020.
(28) Reduce to lowest terms.
Soln. ( 567, 729 ) = 81.
On dividing the numerator and denominator by 81, we get :
= =
(29) The traffic lights at three different road crossing change after everyday 42 sec., 70 sec. and 105 sec. respectively. If they all change simultaneously at 9 : 10 : 00 hours, then at what time will they again change simultaneously?
Soln. Interval of change = L.C.M. of 42, 70, 105 sec. = 210 sec.
So, the lights will again change simultaneously after every 210 seconds
i.e., 3 min. 30 sec.
Hence, next simultaneous change will take place at 9 : 13 : 30 hrs.
(30) Five bells begin to toll together and toll respectively at intervals of 8 sec., 10 sec., 12 sec., 15 sec. and 18 sec. respectively. How many times they will toll together in one hour, excluding the one at the start ?
Soln. Interval of change = L.C.M. of 8, 10, 14, 25, 35 sec. = 1400 sec.
In one hour = 60 min. = 3600 sec., they will toll together = times
= 2 times
(31) The least number of square tiles required to pave the ceiling of a room 15 m long and 9 m 75 cm broad, is :
Soln. The length of a square tile = H.C.F. of 1500, 975 = 75 cm.
The no. of square tiles = = 20 13 = 260
Best of Luck