1.What least value must be assigned to * so that the number 63576*2 is divisible by 8?
a) 1
b) 2
c) 3
d) 4
Explantion:
For a no. to be divisible by 8, the last 3-digits of that no. should be divisible by 8...
Here 6x2 should be divisible by 8...
Try for all possible value from 0 to 9...
x = 3 => 632 which is perfectly divisible by 8...
Ans : 3
2.Which of the following numbers is exactly divisible by 24 ?
a) 35718
b) 63810
c) 537804
d) 3125736
Explanation:
24 = 3 x8, where 3 and 8 co-prime.
Consider option (D),
Sum of digits = (3 + 1 + 2 + 5 + 7 + 3 + 6) = 27, which is divisible by 3.
Also, 736 is divisible by 8.
3125736 is divisible by (3 x 8), i.e., 24.
3.The number nearest to 15207, which is divisible by 467, is:
a) 14342
b) 15211
c) 14944
d) 15411
e) None of these
Explanation:
15207%467=263
467-263=204
15207+204=15411
4. In a division problem, the divisor is twenty times the quotient and five times the remainder. If remainder is 16, the number will be:
a) 3360
b) 336
c) 1616
d) 20516
Explanation:
rem=16, divisor=16*5=80,
quotient=80/20=4;
no.=divisor*quotient+rem
=80*4+16=336
5. If a number is exactly divisible by 85, then what will be the remainder when the same number is divided by 17?
a) 3
b) 1
c) 4
d) 0
Explanation:
number=divisor*quotient+remainder
so 17*5+0;
remainder is 0
6.The least perfect square number which is exactly divisible by 3, 4, 7, 10 and 2 is:
a) 8100
b) 17600
c) 44100
d) None of these
Explanation:
L.C.M. of 3,4,7,10,2 = 420.
Now, 420 =3*4*7*5
420=2^2*5*3*7
15*7=105
⇒420*5=44100
7.(xn+yn) is divisible by (x-y):
a) for all values of n
b) only for even values of n
c) only for odd values of n
d) for no values of n
Ans:
for no values of n
8.The greatest number that will divide 63, 138 and 228 so as to leave the same remainder in each case:
a) 15
b) 20
c) 35
d) 40
Explanation:
HCF of ((138-63),(228-138),(228-63))
HCF of (75,90,165)
75=3*5*5
90=2*3*3*5
165=3*5*11
ans=3*5=15
9.Find the largest number, smaller than the smallest four-digit number, which when divided by 4,5,6 and 7 leaves a remainder 2 in each case.
a) 422
b) 842
c) 12723
d) None of these
Explanation:
lcm of 4,5,6,7 =420
remainder 2 in each case.
therefore 420k+2=1000, 1000 smallest 4 digit number
k=2 840+2=842
10.What is the highest power of 5 that divides 90 x 80 x 70 x 60 x 50 x 40 x 30 x 20 x 10?
Simply if divide all number with 5 than we get reminder as below :
--
90/5 = 18
80/5 = 16
70/5 = 14
60/5 = 12
50/5^2 = 2
40/5 = 8
30/5 = 6
20/5 = 4
10/5 = 2
--
So total highest power will be 9 and 1 add form 50 terms So total will be 10
11.If a and b are natural numbers and a-b is divisible by 3, then a^3-b^3 is divisible by:
a) 3 but not by 9
b) 9
c) 6
d) 27
Explanation:
If a − b is divisible by 3, then a − b = 3k, for some integer k
(a − b)² = (3k)²
a² − 2ab + b² = 9k²
a³ − b³ = (a−b) (a² + ab + b²)
= (a−b) (a² − 2ab + b² + 3ab)
. . . = 3k (9k + 3ab)
. . . = 3k * 3 (3k + ab)
. . . = 9 k(3k+ab)
Since k(3k+ab) is an integer, then 9k(3k+ab) is divisible by 9
Answer: 9
12.What is the greatest positive power of 5 that divides 30! exactly?
a) 5
b) 6
c) 7
d) 8
Explanation:
Only the numbers 5, 10, 15, 20, 25, and 30 have divisors of 5.
And 25 is divisible by 5^2.
So the answer is 5*5*5*5*(5^2)*5 = 5^7
Ans:7
13.What is the Smallest number,which when divided by 7,18,56 and 36,leaves a remainder zero?
a)504
b)392
c)390
d) 1012
Explanation:
l.c.m of 7,18,36 and 56 gives 504
14.What least number must be added to 2010 to obtain a number which is completely divisible by 19?
a)4
b)6
c)8
d) None
Explanation:
2010 divided by 10
Q=15
R=15
=divided - reminder
=19-16=4
Ans: 4
15. What least number must be subtracted from 9400 to get a number exactly divisib 65?
a)40
b)20
c)80
d) none of these
Explanation:
When 94000 is divided by 65
R=40.
Hence if we subtract 40 from 9400 it is exactly divisible by 65
Ans: 40
16.Which smallest number should be added to 86237 to make it exactly divisible by 9?
a) 11
b) 9
c) 10
d) 2
e) None of these
Explanation
When 86237 is divided by 9
R=8
=divided -R
=9-8 =1
Ans:None of these
17.What is the smallest digit which should replace * in the number 296*12 to make it divisible by 12?
a) 1
b) 2
c) 3
d) 4
e) None of these
Explanation
12= 4* 3
if we replace 1 in *
112 is divisble by 3
112 is divisble by 4
so Ans :1
19.If an Integer "K" is divisible by 2,5,and 13 what is the next lagest number that is divisible by all the three given numbers?
a)2k
b)k + 13
c) 2k+13
d)2k + 65
e) k+130
Explanation:
LCM of 2,5,13 is 130
so k=130
2k also the correct answer,but we need to find next largest number
therefore ans is k+130
a) 1
b) 2
c) 3
d) 4
Explantion:
For a no. to be divisible by 8, the last 3-digits of that no. should be divisible by 8...
Here 6x2 should be divisible by 8...
Try for all possible value from 0 to 9...
x = 3 => 632 which is perfectly divisible by 8...
Ans : 3
2.Which of the following numbers is exactly divisible by 24 ?
a) 35718
b) 63810
c) 537804
d) 3125736
Explanation:
24 = 3 x8, where 3 and 8 co-prime.
Consider option (D),
Sum of digits = (3 + 1 + 2 + 5 + 7 + 3 + 6) = 27, which is divisible by 3.
Also, 736 is divisible by 8.
3125736 is divisible by (3 x 8), i.e., 24.
3.The number nearest to 15207, which is divisible by 467, is:
a) 14342
b) 15211
c) 14944
d) 15411
e) None of these
Explanation:
15207%467=263
467-263=204
15207+204=15411
4. In a division problem, the divisor is twenty times the quotient and five times the remainder. If remainder is 16, the number will be:
a) 3360
b) 336
c) 1616
d) 20516
Explanation:
rem=16, divisor=16*5=80,
quotient=80/20=4;
no.=divisor*quotient+rem
=80*4+16=336
5. If a number is exactly divisible by 85, then what will be the remainder when the same number is divided by 17?
a) 3
b) 1
c) 4
d) 0
Explanation:
number=divisor*quotient+remainder
so 17*5+0;
remainder is 0
6.The least perfect square number which is exactly divisible by 3, 4, 7, 10 and 2 is:
a) 8100
b) 17600
c) 44100
d) None of these
Explanation:
L.C.M. of 3,4,7,10,2 = 420.
Now, 420 =3*4*7*5
420=2^2*5*3*7
15*7=105
⇒420*5=44100
7.(xn+yn) is divisible by (x-y):
a) for all values of n
b) only for even values of n
c) only for odd values of n
d) for no values of n
Ans:
for no values of n
8.The greatest number that will divide 63, 138 and 228 so as to leave the same remainder in each case:
a) 15
b) 20
c) 35
d) 40
Explanation:
HCF of ((138-63),(228-138),(228-63))
HCF of (75,90,165)
75=3*5*5
90=2*3*3*5
165=3*5*11
ans=3*5=15
9.Find the largest number, smaller than the smallest four-digit number, which when divided by 4,5,6 and 7 leaves a remainder 2 in each case.
a) 422
b) 842
c) 12723
d) None of these
Explanation:
lcm of 4,5,6,7 =420
remainder 2 in each case.
therefore 420k+2=1000, 1000 smallest 4 digit number
k=2 840+2=842
10.What is the highest power of 5 that divides 90 x 80 x 70 x 60 x 50 x 40 x 30 x 20 x 10?
Simply if divide all number with 5 than we get reminder as below :
--
90/5 = 18
80/5 = 16
70/5 = 14
60/5 = 12
50/5^2 = 2
40/5 = 8
30/5 = 6
20/5 = 4
10/5 = 2
--
So total highest power will be 9 and 1 add form 50 terms So total will be 10
11.If a and b are natural numbers and a-b is divisible by 3, then a^3-b^3 is divisible by:
a) 3 but not by 9
b) 9
c) 6
d) 27
Explanation:
If a − b is divisible by 3, then a − b = 3k, for some integer k
(a − b)² = (3k)²
a² − 2ab + b² = 9k²
a³ − b³ = (a−b) (a² + ab + b²)
= (a−b) (a² − 2ab + b² + 3ab)
. . . = 3k (9k + 3ab)
. . . = 3k * 3 (3k + ab)
. . . = 9 k(3k+ab)
Since k(3k+ab) is an integer, then 9k(3k+ab) is divisible by 9
Answer: 9
12.What is the greatest positive power of 5 that divides 30! exactly?
a) 5
b) 6
c) 7
d) 8
Explanation:
Only the numbers 5, 10, 15, 20, 25, and 30 have divisors of 5.
And 25 is divisible by 5^2.
So the answer is 5*5*5*5*(5^2)*5 = 5^7
Ans:7
13.What is the Smallest number,which when divided by 7,18,56 and 36,leaves a remainder zero?
a)504
b)392
c)390
d) 1012
Explanation:
l.c.m of 7,18,36 and 56 gives 504
14.What least number must be added to 2010 to obtain a number which is completely divisible by 19?
a)4
b)6
c)8
d) None
Explanation:
2010 divided by 10
Q=15
R=15
=divided - reminder
=19-16=4
Ans: 4
15. What least number must be subtracted from 9400 to get a number exactly divisib 65?
a)40
b)20
c)80
d) none of these
Explanation:
When 94000 is divided by 65
R=40.
Hence if we subtract 40 from 9400 it is exactly divisible by 65
Ans: 40
16.Which smallest number should be added to 86237 to make it exactly divisible by 9?
a) 11
b) 9
c) 10
d) 2
e) None of these
Explanation
When 86237 is divided by 9
R=8
=divided -R
=9-8 =1
Ans:None of these
17.What is the smallest digit which should replace * in the number 296*12 to make it divisible by 12?
a) 1
b) 2
c) 3
d) 4
e) None of these
Explanation
12= 4* 3
if we replace 1 in *
112 is divisble by 3
112 is divisble by 4
so Ans :1
19.If an Integer "K" is divisible by 2,5,and 13 what is the next lagest number that is divisible by all the three given numbers?
a)2k
b)k + 13
c) 2k+13
d)2k + 65
e) k+130
Explanation:
LCM of 2,5,13 is 130
so k=130
2k also the correct answer,but we need to find next largest number
therefore ans is k+130
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