Bissoy.com
Doctors Medicines MCQs Q&A

A two-digit positive number is such that the product of its digits is 24. When 18 is added to the number, the digits interchange their places. Which smallest positive number should be subtracted from the given number to make it a perfect square?

A two-digit positive number is such that the product of its digits is 24. When 18 is added to the number, the digits interchange their places. Which smallest positive number should be subtracted from the given number to make it a perfect square? Correct Answer 10

Given:

A two-digit positive number is such that the product of its digits is 24.

When 18 is added to the number, the digits interchange their places.

Concept used:

(A + B)2 = (A - B)2 + 4AB

Calculation:

Let the unit and ten's place digits of the number be P and Q respectively.

The number = 10 × Q + P = 10Q + P

When the digits exchange = 10 × P + Q =10P + Q

According to the question,

PQ = 24      ....(1)

10Q + P + 18 = 10P + Q

⇒ 9P - 9Q = 18

⇒ P - Q = 18/9

⇒ P - Q = 2      ...(2)

⇒ (P - Q)2 = 22

⇒ (P - Q)2 + 4PQ = 22 + 4PQ

⇒ (P + Q)2= 22 + 4 × 24 (From 1)

⇒ (P + Q)2 = 100

⇒ (P + Q) = 10      ....(3)

Now solving (2) and (3),

P = 6, Q = 4

So, the number = 10 × 4 + 6 = 46

Closest perfect square integer less than 46 = 36

Now, the smallest positive number that should be subtracted from 46 to make it a perfect square

⇒ 46 - 36

⇒ 10

∴ 10 should be subtracted from the given number to make it a perfect square.

Related Questions

Given below are three quantities named 1, 2 and 3. Based on the given information, you have to determine the relationship between the three quantities. You should use the given data and knowledge of Mathematics to choose between the possible answer. Quantity 1: Area of the innermost square. A square is made by joining the midpoints of a square of area 200 sq. m. The outermost square is called Square 1, The square inside it is called Square 2. The next square is made by joining midpoints of square 2 and is called Square 3. This process goes on. Square 5 is the innermost square. Quantity 2: A man starts with a speed of 10 km/hr. He increases his speed by 10 km/hr every hour at the start and drives at the same speed for that hour. If he traveled a total distance of 550 km. Find the time Quantity 3: Height of cylinder Its volume is equal to the volume of a cone whose radius is 14 cm and height is 39.3 cm. The radius of the cylinder is 14 cm
The following coding system shows the letter codes for numbers. Excluding the exceptions mentioned below, the digits in the question are to be transformed into letter codes as follows: Digit 0 1 2 3 4 5 6 7 8 9 Letter Code A B C D E F G H I J Exceptions: i. If a number begins with a nonzero even digit, that digit should be coded as N. ii. If a number begins with a nonzero odd digit, that digit should be coded as P. iii. If a number begins and also ends with non-zero even digits, those two digits at the extreme ends should be coded as Q. iv. If a number begins and also ends with nonzero odd digits, those two digits at the extreme ends should be coded as W. v. If a number's middle digit is a multiple of 2 that digit should be coded as Z. Barring these exceptions, the other digits should be codified as per the above-mentioned letter codes. Find the appropriate code for the following number-group 3214067
A two-digit number is such that the product of the digits is 8. If 63 is added to this number, the digits interchange their places. What is the sum of the digits in the number?
A two digit number is such that the product of its digits is 12. When 36 is added to the number, the digits interchange their places. Find the number:
M is the smallest number that has exactly 15 factors. Which of the following is correct? A. M – 23 a perfect square B. Unit digit of M is 9 C. Sum of digits of M is a perfect square D. M is a perfect square
The smallest number which should be subtracted from the smallest number of four digits to make it a perfect square is:
There are two rows, each containing 3 numbers. Some rules are given below to be used for obtaining the resultant number for each row separately. Apply the rules for each row from left to right and answer the question. Rules: i. If an even number is followed by an even number, the first is to be subtracted from the second. ii. If an odd number is followed by an odd number, both are to be added. iii. If an odd number is followed by an even number, the even number is to be subtracted from the square of the odd number. iv. If an even number is followed by an odd number, the odd number is to be subtracted from the even number. Row I: 28, 19, 12 Row II: 13, 7, 32 What is the average of the resultant numbers of the first row and the second row?
There are two rows, each containing 3 numbers. Below are some rules to be used to get the resultant number for each row separately. Use the rule from left to right for each row and find the answer to the question. Rule: 1. If an even number is followed by an even number, then the first number will be subtracted from the second. 2. If an odd number is followed by an odd number, then both will be added. 3. If an odd number is followed by an even number, then the even number will be subtracted from the square of the odd number. 4. If an even number is followed by an odd number, then the odd number will be subtracted from the even number. Row I: 9,11, 34 Row II: 16, 9, 8 What is the difference between the resultant numbers in the first row and the second row
A two-digit number is such that the product of its digits is 14. When 45 is added to this number the digits interchange their position (like 12 + 9 = 21). What is that number.
If 48 is added to a perfect square number, then another perfect square number is obtained. Find how many such pairs of perfect squares exists.