1. The L.C.M. of two numbers is 48. The numbers are in the ratio 2 : 3. Then sum of the number is:
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Correct Ans:40
Explanation:
Let the numbers be 2x and 3x.
Then, their L.C.M. = 6x.
So, 6x = 48 or x = 8.
The numbers are 16 and 24.
Hence, required sum = (16 + 24) = 40.
2. Which of the following has the most number of divisors ?
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Correct Ans:176
Explanation:
99 = 1 x 3 x 3 x 11
101 = 1 x 101
176 = 1 x 2 x 2 x 2 x 2 x 11
182 = 1 x 2 x 7 x 13
So, divisors of 99 are 1, 3, 9, 11, 33, .
99 Divisors of 101 are 1 and 101
Divisors of 176 are 1, 2, 4, 8, 11, 16, 22, 44, 88 and 176
Divisors of 182 are 1, 2, 7, 13, 14, 26, 91 and 182.
Hence, 176 has the most number of divisors.
3. The greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively, is:
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Correct Ans:127
Explanation:
Required number = H.C.F. of (1657 - 6) and (2037 - 5)
= H.C.F. of 1651 and 2032 = 127.
4. The least number, which when divided by 12, 15, 20 and 54 leaves in each case a remainder of 8 is:
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Correct Ans:548
Explanation:
Required number = (L.C.M. of 12, 15, 20, 54) + 8
= 540 + 8
= 548.
5. Find the highest common factor of 36 and 84.
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Correct Ans:12
Explanation:
36 = 2^2 x 3^ 2
84 = 2^2 x 3 x 7
H.C.F. = 2^2 x 3 = 12.
6. Three numbers which are co-prime to each other 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:
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Correct Ans:85
Explanation:
ince the numbers are co-prime, they contain only 1 as the common factor.
Also, the given two products have the middle number in common.
So, middle number = H.C.F. of 551 and 1073 = 29;
First number = (551/29 ) = 19 ; Third number = (1073 / 29) = 37
Therefore, Required sum = (19 + 29 + 37) = 85.
7. The smallest number which when diminished by 7, is divisible 12, 16, 18, 21 and 28 is:
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Correct Ans:1015
Explanation:
Required number = (L.C.M. of 12,16, 18, 21, 28) + 7
= 1008 + 7
= 1015
8. The ratio of two numbers is 3 : 4 and their H.C.F. is 4. Their L.C.M. is:
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Correct Ans:48
Explanation:
Let the numbers be 3x and 4x. Then, their H.C.F. = x. So, x = 4.
So, the numbers 12 and 16.
L.C.M. of 12 and 16 = 48.
9. What will be the least number which when doubled will be exactly divisible by 12, 18, 21 and 30 ?
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Correct Ans:630
Explanation:
L.C.M. of 12, 18, 21 30 2 | 12 - 18 - 21 - 30
----------------------------
= 2 x 3 x 2 x 3 x 7 x 5 = 1260. 3 | 6 - 9 - 21 - 15
----------------------------
Required number = (1260 ÷ 2) | 2 - 3 - 7 - 5
= 630.
10. A, B and C start at the same time in the same direction to run around a circular stadium. A completes a round in 252 seconds, B in 308 seconds and c in 198 seconds, all starting at the same point. After what time will they again at the starting point ?
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Correct Ans:46 minutes and 12 seconds
Explanation:
L.C.M. of 252, 308 and 198 = 2772.
So, A, B and C will again meet at the starting point in 2772 sec. i.e., 46 min. 12 sec.
11. The least number which when divided by 5, 6 , 7 and 8 leaves a remainder 3, but when divided by 9 leaves no remainder, is:
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Correct Ans:1683
Explanation:
L.C.M. of 5, 6, 7, 8 = 840.
Required number is of the form 840k + 3
Least value of k for which (840k + 3) is divisible by 9 is k = 2.
Required number = (840 x 2 + 3) = 1683.
12. The least number which should be added to 2497 so that the sum is exactly divisible by 5, 6, 4 and 3 is:
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Correct Ans:23
Explanation:
L.C.M. of 5, 6, 4 and 3 = 60.
On dividing 2497 by 60, the remainder is 37.
Number to be added = (60 - 37) = 23.
13. The product of two numbers is 2028 and their H.C.F. is 13. The number of such pairs is:-
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Correct Ans:2
Explanation:
Let the two numbers be 13a and 13b respectively.
Then, 13a x 13b = 2028
=>ab = 2028 / (13*13) = 12.
Now, the co-primes with product 12 are (1, 12) and (3, 4).
[Note: Two integers a and b are said to be coprime or relatively prime if they have no common positive factor other than 1 or, equivalently, if their greatest common divisor is 1 ]
So, the required numbers are (13 x 1, 13 x 12) and (13 x 3, 13 x 4). Clearly, there are 2 such pairs.
14. The product of two numbers is 4107. If the H.C.F. of these numbers is 37, then the greater number is:
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Correct Ans:111
Explanation:
Let the numbers be 37a and 37b.
Then, 37a x 37b = 4107
=> ab = 3.
Now, co-primes with product 3 are (1, 3).
So, the required numbers are (37 x 1, 37 x 3) i.e., (37, 111).
Therefore, Greater number = 111.
15. The greatest number of four digits which is divisible by 15, 25, 40 and 75 is:
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Correct Ans:9600
Explanation:
Greatest number of 4-digits is 9999.
L.C.M. of 15, 25, 40 and 75 is 600.
On dividing 9999 by 600, the remainder is 399.
Required number (9999 - 399) = 9600.
16. Let N be the greatest number that will divide 1305, 4665 and 6905, leaving the same remainder in each case. Then sum of the digits in N is:
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Correct Ans:4
Explanation:
N = H.C.F. of (4665 - 1305), (6905 - 4665) and (6905 - 1305)
= H.C.F. of 3360, 2240 and 5600 = 1120.
Sum of digits in N = ( 1 + 1 + 2 + 0 ) = 4
17. Six bells commence tolling together and toll at intervals of 2, 4, 6, 8 10 and 12 seconds respectively. In 30 minutes, how many times do they toll together ?
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Correct Ans:16
Explanation:
L.C.M. of 2, 4, 6, 8, 10, 12 is 120.
So, the bells will toll together after every 120 seconds(2 minutes).
In 30 minutes, they will toll together 30/2 + 1 = 16 times
18. Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.
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Correct Ans:4
Explanation:
Required number = H.C.F. of (91 - 43), (183 - 91) and (183 - 43)
H.C.F. of 48, 92 and 140 = 4.
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