The difference between a 2-digit number and the number obtained by interchanging the positions of the digits is 54. Consider the following statements: 1. The sum of the two digits of the number can be determined only if the product of the two digits is known. 2. The difference between the two digits of the number can be determined. Which of the above statements is/are correct?
2 only
Calculation:
Let us assume the number has y in units place and x in tens place.
So, the number can be written as 10x + y
Now,
if we interchange the digits ⇒ x in units place and y in tens place
The number obtained = 10y + x
As given in the question we can write
(10x + y) - (10y + x) = 54
⇒ 9x - 9y = 54 = 9(x - y) = 54
⇒ x - y = 6
⇒ x = y + 6
Here y can't be 0 as after interchanging the digits the number formed will not be a 2-digit number. So the possible outcomes are
If y = 1, x = 7 and the numbers are 71 and 17
If y = 2, x = 8 and the numbers are 82 and 28
If y = 3, x = 9 and the numbers are 93 and 39
Statement 1 is incorrect as by knowing the products of digit is not the only way. If their ratios are given then also we can find the numbers.
Statement 2 is correct as the difference of the digits is obtained by the equation x - y = 6.
Hence, option 2 is correct.
Important Questions from Integers
Find the number of integers between $1$ and $150$ (inclusive) having $7$ as one of the digits but which are not divisible by $7$.
Integers are listed from 700 to 1000. In how many integers is the sum of the digits 10 ?
Using 2, 2, 3, 3, 3 as digits, how many distinct numbers greater than 30000 can be formed ?
Consider the following statements :
1. The sum of 5 consecutive integers can be 100.
2 The product of three consecutive natural numbers can be equal to their sum.
Which of the above statements is/are correct ?
When a certain number is multiplied by 7, the product entirely comprises ones only (1111....). What is the smallest such number?
