Question

3. Find the greatest number of 6 digits which on dividing by 15, 18, 21, 24 and 27 leaves 10,13,16,19and22
and 15 as remainder.
and 22 as remainder.
-. What is the smallest number which when divided by 24, 36 and 54 gives 5 as remainderea
f16 20 and 25 seconds respectively. If they start ringing
lust​

Answer 1

Given : greatest number of 6 digits which on dividing by 15, 18, 21, 24 and 27 leaves 10,13,16,19 and 22

To find : Number

Solution:

Let say number = N

N = 15A  + 10  = 15(A + 1) - 5  =>  N + 5 = 15(A + 1)

N = 18B + 13  = 18(B + 1) - 5   => N + 5 = 18(B + 1)

N = 21C + 16  = 21(C + 1) - 5  => N + 5 = 21(C + 1)

N = 24D + 19 = 24(D + 1) - 5 => N + 5 = 24(D + 1)

N = 27E + 22 = 27(E + 1) - 5  => N + 5 = 27(E + 1)

=> N + 5 is a  product of  LCM of   15 , 18 , 21 , 24  & 27

15 = 3 * 5

18 = 2 * 3 * 3

21 = 3 * 7

24 = 2 * 2 * 2 * 3

27 = 3 * 3 * 3

LCM = 2 * 2 * 2 * 3 * 3 * 3 * 5 * 7  =  7560

Largest 6 digit number = 999999

999999 = 7560 * 132 + 2079

=> N + 5 =  7560 * 132

=> N + 5 = 997920

=> N = 997915

997915 is the greatest 6 digit number  which on dividing by 15, 18, 21, 24 and 27 leaves 10,13,16,19 and 22

N - 5 is LCM of 24 , 36 & 54

24 = 2 * 2 * 2 * 3

36 = 2 * 2 * 3 * 3

54 = 2 * 3 * 3 * 3

LCM = 216

=> N = 221   is smallest number which when divided by 24, 36 and 54 gives 5 as remainder

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