Number Problems
Q1
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?
Ans:
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.
Q2
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?
Ans:
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.
Q3
The product of two numbers is 9375 and the quotient, when the larger one is divided by the smaller, is 15. The sum of the numbers is?
Ans:
Let the numbers be x and y.
Then, xy = 9375 and x/y=15
xy/(x/y) = 9375/15
y = 25. y2 = 625.
x = 15y = (15 x 25) = 375.
Sum of the numbers = x + y = 375 + 25 = 400.
Q4
The product of two numbers is 120 and the sum of their squares is 289. The sum of the number is?
Ans:
Let the numbers be x and y.
Then, xy = 120 and x2 + y2 = 289.
(x + y)2 = x2 + y2 + 2xy = 289 + (2 x 120) = 529
x + y = 529 = 23.
Q5
What is the sum of two consecutive even numbers, the difference of whose squares is 84?
Ans:
Let the numbers be x and x + 2.
Then, (x + 2)2 - x2 = 84
4x + 4 = 84
4x = 80
x = 20.
The required sum = x + (x + 2) = 2x + 2 = 42.
Q6
The sum of two number is 25 and their difference is 13. Find their product?
Ans:
Let the numbers be x and y.
Then, x + y = 25 and x - y = 13.
4xy = (x + y)2 - (x- y)2
= (25)2 - (13)2
= (625 - 169)
= 456
xy = 114.