# problems on numbers

6. The sum of the digits of a two-digit number is 15 and the difference between the digits is 3. What is the two-digit number?
 A. 69 B. 78 C. 96 D. Cannot be determined E. None of these

Explanation:

Let the ten’s digit be x and unit’s digit be y.

Then, x + y = 15 and xy = 3   or   yx = 3.

Solving x + y = 15   and   xy = 3, we get: x = 9, y = 6.

Solving x + y = 15   and   yx = 3, we get: x = 6, y = 9.

So, the number is either 96 or 69.

Hence, the number cannot be determined.

7. The sum of the squares of three numbers is 138, while the sum of their products taken two at a time is 131. Their sum is:
 A. 20 B. 30 C. 40 D. None of these

Explanation:

Let the numbers be a, b and c.

Then, a2 + b2 + c2 = 138 and (ab + bc + ca) = 131.

(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca) = 138 + 2 x 131 = 400.

(a + b + c) = 400 = 20.

8. A number consists of two digits. If the digits interchange places and the new number is added to the original number, then the resulting number will be divisible by:
 A. 3 B. 5 C. 9 D. 11

Explanation:

Let the ten’s digit be x and unit’s digit be y.

Then, number = 10x + y.

Number obtained by interchanging the digits = 10y + x.

(10x + y) + (10y + x) = 11(x + y), which is divisible by 11.

9. In a two-digit, if it is known that its unit’s digit exceeds its ten’s digit by 2 and that the product of the given number and the sum of its digits is equal to 144, then the number is:
 A. 24 B. 26 C. 42 D. 46

Explanation:

Let the ten’s digit be x.

Then, unit’s digit = x + 2.

Number = 10x + (x + 2) = 11x + 2.

Sum of digits = x + (x + 2) = 2x + 2.

(11x + 2)(2x + 2) = 144

22x2 + 26x – 140 = 0

11x2 + 13x – 70 = 0

(x – 2)(11x + 35) = 0

x = 2.

Hence, required number = 11x + 2 = 24.

10. Find a positive number which when increased by 17 is equal to 60 times the reciprocal of the number.
 A. 3 B. 10 C. 17 D. 20

Explanation:

Let the number be x.

 Then, x + 17 = 60 x

x2 + 17x – 60 = 0

(x + 20)(x – 3) = 0

x = 3.