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Examples(2)
(11) Greatest number which can divide 327, 352, 200 leaving respectively 12, 7, 5 as remainders, is :
Soln. Required number = H.C.F. of ( 327 – 12 ), ( 352 – 7 ), ( 200 – 5 )
= H.C.F. of 315, 345, 195
= 15
(12) The largest number which divides 223, 559 and 433 to leave the same remainder in each case, is :
Soln. Required number = H.C.F. of ( 559 – 223 ), ( 559 – 433 ), ( 433 – 223 )
= H.C.F. of 336, 126, 210
= 42
(13) The largest number which divides 157, 67 and 277 to leave the same remainder in each case, is :
Soln. Required number = H.C.F. of ( 277 – 157 ), ( 277 – 67 ), ( 157 – 67 )
= H.C.F. of 120, 210, 90
= 30
(14) If p is a prime number, then ( p, p + 1 ) = ……. & [ p, p + 1 ] = …….
Soln. Since p is a prime number, ( p, p + 1 ) = 1 & [ p, p + 1 ] = p · ( p + 1 )
(15) Find the least number which is divisible by 10, 16, 24 and 36.
Soln. Required number = [ 10, 16, 24, 36 ]
= 720
(16) Find the least number which when divided by 12, 15, 20 and 36 leaves the same remainder 2 in each case.
Soln. Required number = [12, 15, 20, 36 ] + 2 = 180 + 2 = 182.
(17) Find the least number which when divided by 20, 28, 15 and 27 leaves the same remainder 14 in each case.
Soln. Required number = [20, 28, 15, 27 ] + 14 = 3780 + 14 = 3794.
(18) The least number which when divided by 3, 5, 6 and 7 leaves a remainder 2, but when divided by 8 leaves no remainder, is :
Soln. L. C. M. of 3, 5, 6, 7 = 210.
Required number is of the form of 210 k + 2.
If k = 1, then we have 212. Which is not divisible by 8.
If k = 2, then we have 422. Which is not divisible by 8.
If k = 3, then we have 632. Which is divisible by 8.
Required number = 632
(19) The least number which when divided by 4, 6, 8 and 9 leaves a remainder 3, but when divided by 11 leaves no remainder, is :
Soln. L. C. M. of 4, 6, 8, 9 = 72.
Required number is of the form of 72 k + 3.
If k = 1, then we have 75. Which is not divisible by 11.
If k = 2, then we have 147. Which is not divisible by 11.
If k = 3, then we have 219. Which is not divisible by 11.
If k = 4, then we have 291. Which is not divisible by 11.
If k = 5, then we have 363. Which is divisible by 11.
Required number = 363
(20) Find the lowest number which when subtracted from 5000, is exactly divisible by 9, 30 and 25.
Soln. L.C.M. of 9, 30 and 25 = 450.
If we divide 5000 by 450, we get the remainder = 50
i.e. If 50 is subtracted from 5000, then the no. so obtained is divisible by 9, 30 & 25.
