Ques 11 : Choose the correct answer.

The L.C.M. of two numbers is 4800 and their G.C.M. is 160. If one of the numbers is 480, then the other number is:

Option 2 : 1800

Option 3 : 2200

Option 4 : 2600

Option 5 : None of these

Answer : C

Explanation :

Let n be the required number.

Since, HCF x LCM = product of two numbers.

=>4800 x 160 = 480 x n.

=>n=(4800 x 160)/480.

=>n=1600.

Ques 12 : Choose the correct answer.

The L.C.M. of two numbers is 140. If their ratio is 2:5, then the numbers are:

Option 2 : 28,7

Option 3 : 8,70

Option 4 : 8,40

Option 5 : None of these

LCM = 140

Let HCF be x

Then the 2 nos. ll be 2x nd 5x

Then Prod. of 2 nos. = LCM*HCF

2x * 5x = 140x

10x = 140

x = 14

2 nos. ll be 2*14 nd 5*14 => 28, 70

Ans : 28, 70

Ques 13 : Choose the correct answer.

If a number is exactly divisible by 85, then what will be the remainder when the same number is divided by 17?

Option 1 : 3

Option 2 : 1

Option 3 : 4

zero 17*5=85 so remainder =0

Ques 14 : Choose the correct answer.

The least perfect square number which is exactly divisible by 3, 4, 7, 10 and 12 is:

Option 1 : 8100

Option 2 : 17600

Option 4 : None of these

LCM (3, 4, 5, 6 ,8) = 120

120 = 2 x 2 x 2 x 3 x 5.

As 2,3 and 5 are not in pair in LCM’s factor so we need to multiply 120 by 5 and 3,2 to make it a perfect square.

120 x 2 x 5 x 3 = 3,600.

∴ 3600 is the least perfect square divisible by 3, 4, 5, 6 and 8.

Ques 15 : Choose the correct answer.

(xn+yn) is divisible by (x-y):

Option 1 : for all values of n

Option 2 : only for even values of n

Option 3 : only for odd values of n

Ques 16 : Choose the correct answer.

The greatest number that will divide 63, 138 and 228 so as to leave the same remainder in each case:

Option 2 : 20

Option 3 : 35

Option 4 : 40

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

Ques 17 : Choose the correct answer.

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.

Option 1 : 422

Option 3 : 12723

Option 4 : None of these

Take LCM of 4,5,6,7. It is 420

BUt the no must leave remainder 2 in each case, so the no is of the form: 420k + 2.

The smallest 4-digit no is 1000. So keeping k=0,1,2,3....

We get that the largest no smaller than the smallest 4 -digit no is 842

Ques 18 : Choose the correct answer.

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?

Option 2 : 12

Option 3 : 14

Option 4 : None of these

This number can be factorized into (18*5)*(16*5)*(14*5)*(12*5)*(2*5*5)*(8*5)*(6*5)*(4*5)*(2*5)

Total number of 5’s in this factor=10 is the highest power of 5 that divides this number.

Ques 19 : Choose the correct answer.

If a and b are natural numbers and a-b is divisible by 3, then a3-b3 is divisible by:

Option 1 : 3 but not by 9

Option 3 : 6

Option 4 : 27

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

Ques 20 : Choose the correct answer.

What is the greatest positive power of 5 that divides 30! exactly?

Option 1 : 5

Option 2 : 6

Option 4 : 8

5,10,15,20,25 & 30(When we write 30! as 1*2*3*4*5........*28*29*30(

Are having 7 , 5s as factors.

The L.C.M. of two numbers is 4800 and their G.C.M. is 160. If one of the numbers is 480, then the other number is:

**Option 1 : 1600**Option 2 : 1800

Option 3 : 2200

Option 4 : 2600

Option 5 : None of these

Answer : C

Explanation :

Let n be the required number.

Since, HCF x LCM = product of two numbers.

=>4800 x 160 = 480 x n.

=>n=(4800 x 160)/480.

=>n=1600.

Ques 12 : Choose the correct answer.

The L.C.M. of two numbers is 140. If their ratio is 2:5, then the numbers are:

**Option 1 : 28,70**Option 2 : 28,7

Option 3 : 8,70

Option 4 : 8,40

Option 5 : None of these

LCM = 140

Let HCF be x

Then the 2 nos. ll be 2x nd 5x

Then Prod. of 2 nos. = LCM*HCF

2x * 5x = 140x

10x = 140

x = 14

2 nos. ll be 2*14 nd 5*14 => 28, 70

Ans : 28, 70

Ques 13 : Choose the correct answer.

If a number is exactly divisible by 85, then what will be the remainder when the same number is divided by 17?

Option 1 : 3

Option 2 : 1

Option 3 : 4

**Option 4 : 0**zero 17*5=85 so remainder =0

Ques 14 : Choose the correct answer.

The least perfect square number which is exactly divisible by 3, 4, 7, 10 and 12 is:

Option 1 : 8100

Option 2 : 17600

**Option 3 : 44100**Option 4 : None of these

LCM (3, 4, 5, 6 ,8) = 120

120 = 2 x 2 x 2 x 3 x 5.

As 2,3 and 5 are not in pair in LCM’s factor so we need to multiply 120 by 5 and 3,2 to make it a perfect square.

120 x 2 x 5 x 3 = 3,600.

∴ 3600 is the least perfect square divisible by 3, 4, 5, 6 and 8.

Ques 15 : Choose the correct answer.

(xn+yn) is divisible by (x-y):

Option 1 : for all values of n

Option 2 : only for even values of n

Option 3 : only for odd values of n

**Option 4 : for no values of n**Ques 16 : Choose the correct answer.

The greatest number that will divide 63, 138 and 228 so as to leave the same remainder in each case:

**Option 1 : 15**Option 2 : 20

Option 3 : 35

Option 4 : 40

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

Ques 17 : Choose the correct answer.

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.

Option 1 : 422

**Option 2 : 842**Option 3 : 12723

Option 4 : None of these

Take LCM of 4,5,6,7. It is 420

BUt the no must leave remainder 2 in each case, so the no is of the form: 420k + 2.

The smallest 4-digit no is 1000. So keeping k=0,1,2,3....

We get that the largest no smaller than the smallest 4 -digit no is 842

Ques 18 : Choose the correct answer.

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?

**Option 1 : 10**Option 2 : 12

Option 3 : 14

Option 4 : None of these

This number can be factorized into (18*5)*(16*5)*(14*5)*(12*5)*(2*5*5)*(8*5)*(6*5)*(4*5)*(2*5)

Total number of 5’s in this factor=10 is the highest power of 5 that divides this number.

Ques 19 : Choose the correct answer.

If a and b are natural numbers and a-b is divisible by 3, then a3-b3 is divisible by:

Option 1 : 3 but not by 9

**Option 2 : 9**Option 3 : 6

Option 4 : 27

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

Ques 20 : Choose the correct answer.

What is the greatest positive power of 5 that divides 30! exactly?

Option 1 : 5

Option 2 : 6

**Option 3 : 7**Option 4 : 8

5,10,15,20,25 & 30(When we write 30! as 1*2*3*4*5........*28*29*30(

Are having 7 , 5s as factors.

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