Question 1

The number 6 2n - 1, where n is any positive integer, is always divisible by?

Accepted Answer

62 -1 = 35, which is divisible by both 5 and 7.

Question 2

Which of the following numbers is not a square of any natural number?

Accepted Answer

A number that ends in 2 , 3 , 7 or 8 is never a perfect square.

Question 3

Of the three numbers, the first is twice the second and is half of the third. if the average of three number is 56. the smallest number is?

Accepted Answer

Let the second number be N. Then, first number = 2N and third number = 4N ∴ 2N + N+ 4N/3 = 56 ⇒ 7N = 3 x 56 ∴ N = 3 x 56/7 = 24 So, the smallest number is 24.

Question 4

The number x is exactly divisible by 5 and the remainder obtained on dividing the number y by 5 is 1. what remainder will be obtained when (x + y) is divided by 5?

Accepted Answer

Let x/5 = p and let y when divided by 5 gives q as quotient and 1 as remainder. Then, y = 5q + 1 Now x = 5p and y = 5q + 1 ∴ x + y = 5p + 5q + 1 = 5(p + q) + 1 So, required remainder = 1

Question 5

If the unit digit in the product 75? x 49 x 867 x 943 be 1, then the value of? is

Accepted Answer

Let the value for ? be N So N x 9 x 7 x 3 = a number with unit digit 1. Clearly, the minimum value of N is 9.

Question 6

The sum of three numbers is 132, If the first number be twice the second and third number be one-third of the first, then the second number is?

Accepted Answer

Let second number be 3N, then, first one is 6N and the third one is 2N. From question 3N + 6N + 2N = 132 ⇒ 11N = 132 ⇒ N = 12 ∴ Second number = 3N = 3 x 12 = 36

Question 7

A certain number of two digits is three times the sum of its digits and if 45 be added to it, then digits are reversed. The number is?

Accepted Answer

Let unit digit = x and ten's digit = y 3(x + y)= 10y + x .....(i) 10y + x + 45 = 10x + y ....(ii) 2x - 7y = 0.9x -9y = 45 or x - y = 5 .... (iii) Solving these equations, we get x=7, y=2 ∴ Required number = 27

Question 8

A fraction becomes 4 when 1 is added to both the numerator and denominator, it becomes 7 When 1 is subtract from both the numerator and denominator. The numerator of the given fractions is?

Accepted Answer

Let the required fraction be p/q. Then, (p +1) / (q +1) = 4 ⇒ p - 4q= 3 ....(i) And, (p - 1) / (q -1) = 7 ⇒ p - 7q = -6 ....(ii) Solving these equation we get p = 15, q=3

Question 9

The sum of squares of two numbers is 68 and the squares of their difference is 36. The product of the two number is?

Accepted Answer

Let the number be a and b . Then a2 + b2 = 68 ....(i) and (a - b)2 = 36 Now, (a - b)2 = 36 ....(ii) ⇒ a2 + b2 - 2ab = 36 ⇒ 68 - 2ab = 36 ⇒ 2ab = 32 ⇒ ab = 16 .

Question 10

There are two numbers such that the sum of twice the first and thrice the second is 18, while the sum of thrice the first and twice the second is 17. The larger of the two is?

Accepted Answer

Let the numbers be x and y , Then, 2x + 3y = 18 ....(i), 3x + 2y = 17 ....(ii) Solving, we get x =3 , y= 4 ∴ Largest number = 4

Question 11

Of the three numbers, the sum of first two is 45, the sum of the second and the third is 55 and the sum of the third and thrice the first is 90. The third number is?

Accepted Answer

Let the number be x, y, z Then, x + y = 45, y + z = 55, z + 3x = 30 Now, y = (45 - x) and z = 55 - y ⇒ z = 55 - (45 - x ) = 10 + x ∴ 10 + x + 3x = 90 ⇒ x = 20 So, third number = z = 10 + x = 30

Question 12

A number when divided by 5 leaves a remainder 3. What is the remainder when the square of the same number is divided by 5?

Accepted Answer

The number is of the form (5x + 3 ) where x is an integer ∴ [(5x + 3)2] / 5 = ( 25x2 + 30x + 9) / 5 = 25x2/ 5 + 30x/5 + (5 +4)/5 ∴ The remainder is 4.

Question 13

In a question divisor is 2/3 of the dividend and 2 times the remainder. If the remainder is 5, find the dividend.?

Accepted Answer

According to the question Divisor = 2/3 x dividend and Divisor = 2 x remainder or 2/3 x dividend = 2 x 5 ∴ Dividend = 2 x 5 x 3 / 2 = 15

Question 14

How many figures (digits) are required to number a book containing 200 pages?

Accepted Answer

Number of one digit pages from 1 to 9 = 9 Number of two digit pages from 10 to 99 = 90 Number of three digit pages from 100 to 200 = 101 ∴ Total number of required figures = 9 x 1 + 90 x 2 + 101 x 3 = 492

Question 15

The digit in the units place of a number is equal to the digit in the tens place of half of that number and the digit in the tens place of that number is less than the digit in units place of half of the number by 1, If the sum of the digits of the …

Accepted Answer

Let 1/2 of the no. = 10x + y and the no. = 10v + w From the given conditions, w= x and v = y-1 Thus the no. = 10 (y-1) + x ∴ 2(10x + y ) = 10 (y-1) + x ⇒ 8y - 19x = 10 ...(i) v + w = 7 ⇒ y-1 + x = 7 ∴ x + y = 8 Solving equations (i) and (ii) , we get x = 2 and y = 6 ∴ From equations (A) Number = 10 (y - 1) + x = 52.

Question 16

A two-digit number is seven times the sum of its digits, If each digit is increased by 2, the number thus obtained is 4 more than six times the sum of its digits, Find the number?

Accepted Answer

Let the two-digit number be 10x + y 10x + y = 7(x + y) ⇒ x = 2y ...(i) 10(x +2 ) + (y + 2) = 6(x + y + 4) + 4 or 10x + y + 22 = 6x + 6y + 28 ⇒ 4x - 5y = 6 ...(ii) Solving equations (i) and (ii) We get x = 4 and y = 2

Question 17

If a number is decreased by 4 and divided by 6 the result is 9. What would be the result if 3 is subtracted from the number and then it is divided by 5?

Accepted Answer

Let the number be x. ∴ (x - 4)/6 = 9 ⇒ x = 58 Again (x - 3)/5 = (58 - 3)/5 = 11

Question 18

The numbers x, y, z are such that xy = 96050 and xz = 95625 and y is greater than z by one. Find out the number z?

Accepted Answer

xy = 96050 ...(i) and xz = 95625 ...(ii) and y - z = 1 ... (iii) Dividing (i) by (ii) we get y/z = 96050 / 95625 = 3842 / 3825 = 226 / 225 ... (iv) Combining (iii) and (iv) we get z = 225.

Question 19

The ratio of two number is 3:2, If 10 and the sum of the two numbers are added to their product, square of sixteen is obtained. What could be the smaller number?

Accepted Answer

Let the two numbers be 3N and 2N According to the question. 10 + (3N +2N ) + (3N x 2N) = 162 ⇒ 6N2 + 5N - 246 = 0 ⇒ 6N2 + 41N - 36N - 246 = 0 ⇒ N(6N + 41 ) - 6(6N + 41) = 0 ⇒ (6N + 41 ) (N - 6) = 0 ∴ N = 6 or -41/6 (But -ve value cannot be accepted ) ∴ Smaller number = 2N = 2 x 6 = 12 .

Question 20

In a two digit number, the digital unit place is 1 more than twice of the digit at ten's place. If the digit at unit's and ten's place be interchanged, then the difference between the new number and original number is less than 1 to that of original…

Accepted Answer

Let the original number be 10p + q. So from question, q = 2p + 1 ...(i) and (10q + p) - (10p + q ) = (10p + q) - 1 ⇒ 9q - 9p = 10p + q -1 ⇒ 19p - 8q = 1...(ii) Putting the value of (i) in equation (ii) we get 19p - 8(2p + 1) = 1 ⇒ 19p - 16p - 8 = 1 ⇒ 3p = 9 ⇒ p = 3 So, q = 2 x 3 + 1 = 7 ∴ Original number = 10 x 3 + 7 = 37

Number System Questions