16. Let x be the least number, which when divided by 5, 6, 7 and 8 leaves a remainder of 3 in each case but when divided by 9 leaves no remainder. The sum of the digits of x is
a) 21 b) 18 c) 22 d) 24
Correct Answer: (b)
LCM of 5, 6, 7 and 8 = 840
2 5, 6, 7, 8 — divide by 2
5, 3, 7, 4
∴ LCM = 2 × 5 × 3 × 7 × 4 = 840
Required number = 840x + 3
which is divisible by 9 for a certain least value of x.
Now, 840x + 3 = 93x × 9 + 3x + 3
3x + 3, is divisible by 9 for x = 2
Required number = 840 × 2 + 3
= 1680 + 3 = 1683
∴ Sum of digits = 1 + 6 + 8 + 3 = 18
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17. Three coprime numbers are such that the product of the first two is 551 and that of the last two is 1073. The sum of the three numbers is:
a) 75 b) 85 c) 81 d) 89
Correct Answer: (b)
Let the numbers be x, y and z which are prime to one another.
Now, xy = 551 yz = 1073
∴ y = HCF of 551 and 1073
y = 29
x = 551/29 = 19
and z = 1073/29 = 37
∴Sum = 19 + 29 + 37 = 85
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18. The greatest four-digit number which is exactly divisible by each one of the numbers 12, 18, 21 and 28 is
a) 9828 b) 9882 c) 9288 d) 9928
Correct Answer: (a)
Using Rule 1,
1st number × 2nd number = L.C.M. × H.C.F
2 12, 18, 21, 28 — divide by 2
2 6, 9, 21, 14 — divide by 2
3 3, 9, 21, 7 — divide by 3
7 1, 3, 7, 7 — divide by 7
1, 3, 1, 1
LCM = 2 × 2 × 3 × 3 × 7= 252
The largest 4-digit number = 9999
Then 9999/252 = 171
∴ Required number = 9999 – 171 = 9828
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20. The number between 4000 and 5000 that is divisible by each of 12, 18, 21 and 32 is
