To solve the problem of finding the number of pairs of natural numbers \( m \) and \( n \) (where \( m > n \)) such that the difference of their squares is 72, we start by using the formula for the difference of squares:
\(m^2 - n^2 = (m - n)(m + n) = 72\)
Here, we need to find integer solutions for pairs \( (m, n) \) where the product of \((m - n)\) and \((m + n)\) equals 72.
Step-by-step solution:
- Identify factor pairs of 72. The factor pairs of 72 are:
- (1, 72)
- (2, 36)
- (3, 24)
- (4, 18)
- (6, 12)
- (8, 9)
- Since \( m \) and \( n \) are natural numbers, both \( (m-n) \) and \( (m+n) \) must be positive integers.
- Substitute each pair of factors into the equations:
- \( m - n = 1 \) and \( m + n = 72 \):
- Adding the two equations: \( 2m = 73 \), which does not give an integer \( m \).
- \( m - n = 2 \) and \( m + n = 36 \):
- Adding the two equations: \( 2m = 38 \), thus \( m = 19 \).
- Subtracting the two equations: \( 2n = 34 \), thus \( n = 17 \).
- This pair \((m, n) = (19, 17)\) is valid.
- \( m - n = 3 \) and \( m + n = 24 \):
- Adding the two equations: \( 2m = 27 \), which does not give an integer \( m \).
- \( m - n = 4 \) and \( m + n = 18 \):
- Adding the two equations: \( 2m = 22 \), thus \( m = 11 \).
- Subtracting the two equations: \( 2n = 14 \), thus \( n = 7 \).
- This pair \((m, n) = (11, 7)\) is valid.
- \( m - n = 6 \) and \( m + n = 12 \):
- Adding the two equations: \( 2m = 18 \), thus \( m = 9 \).
- Subtracting the two equations: \( 2n = 6 \), thus \( n = 3 \).
- This pair \((m, n) = (9, 3)\) is valid.
- \( m - n = 8 \) and \( m + n = 9 \):
- Adding the two equations: \( 2m = 17 \), which does not give an integer \( m \).
- \( m - n = 1 \) and \( m + n = 72 \):
From the above calculations, the valid pairs are: (19, 17), (11, 7), and (9, 3). Therefore, there are 3 pairs of natural numbers that satisfy the given condition.
Similar Questions
- Let \(x\) and \(y\) be natural numbers, each less than 20, such that \(x\), \(y\), \(x+y\) and \(x-y\) are prime numbers. How many such combinations of \((x, y, x+y, x - y)\) are possible?
- A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question : Is \(r^n\) less than 1, where \(r\) is a real number and \(n\) is a natural number ?
Statement-I : \(0 < r^2 < 1\)
Statement-II : \(0 < r^3 < 2\)
Which one of the following is correct in respect of the above Question and the Statements ? - A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question : Is \((0.5)^n + (0.5)^{-n}\) always greater than 2, where \(n\) is an integer ?
Statement-I : \(n\) is an even integer
Statement-II : \(n\) is negative
Which one of the following is correct in respect of the above Question and the Statements ? - A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question : Let XYZ be a 3-digit number and the difference between XYZ and ZYX is equal to PQR. Is (P + R) equal to Q ?
Statement-I : P = 3
Statement-II : R = 6
Which one of the following is correct in respect of the above Question and the Statements ? - A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.
Question : Is \((p^2 + q^2)\) always composite number, where \(p\) and \(q\) are different prime numbers ?
Statement-I : \((p - q)\) is an odd integer
Statement-II : \((p + q)\) is an odd integer
Which one of the following is correct in respect of the above Question and the Statements ? - What are the last three digits in the multiplication of
\(4321012345 \times 98766789\)? What is the sum of the largest and the smallest 4-digit numbers made by using single digit prime numbers (without repetition)?
If \(11x + 5y\) is a prime number where \(x, y\) are natural numbers then what is the minimum value of \((x + y)\) ?
A 4-digit number \(N\) has exactly 15 distinct divisors. What is the total number of distinct divisors of \(N^2\)?
- Consider the following statements in respect of \(p = n(n + 1)(n+2)(n + 3) + 1\), where \(n\) is a natural number :
I. \(p\) is always odd
II. \(p\) is a perfect square
Which of the statements given above is/are correct?
Important Questions from Number System
What is the value of 1 2 + 2 2 + 3 2 + ......21 2 ?
Which sequence is correct to represent the hierarchical chain of number system?
(Where N - Natural Numbers
W - Whole Numbers
Q - Rational Numbers
Z - Integers)
What must be added to 45680 to make it exactly divisible by 9?
How many zeroes are there at the end of the following product?
1 x 5 x 10 x 15 x 20 x 25 x 30 x 35 x 40 x 45 x 50 x 55 x 60
Let XYZ be a three-digit number, where (x + y + Z) is not a multiple of 3. Then (XYZ + YZX + ZXY) is not divisible by
