Finding the Other Number Using HCF and LCM

The question asks us to find the other number when we are given the Highest Common Factor (HCF), the Least Common Multiple (LCM) of two numbers, and one of the numbers. There is a fundamental relationship that connects the HCF and LCM of two positive integers to the product of these two numbers.

Relationship Between HCF, LCM, and Product of Two Numbers

For any two positive integers, say 'a' and 'b', the product of their HCF and LCM is equal to the product of the numbers themselves. This can be expressed by the formula:

HCF(a, b) × LCM(a, b) = a × b

Applying the Formula to the Problem

In this problem, we are given:

• HCF of the two numbers = 8
• LCM of the two numbers = 2520
• One of the numbers = 56
• Let the other number be 'x'.

Using the formula, we can write the equation:

HCF × LCM = One Number × Other Number

Substituting the given values:

\( 8 \times 2520 = 56 \times x \)

Solving for the Other Number

To find the value of 'x', we need to isolate 'x' in the equation. We can do this by dividing both sides of the equation by 56:

\( x = \frac{8 \times 2520}{56} \)

Now, we can perform the calculation. We can simplify the fraction first.

We know that 56 is \( 8 \times 7 \). So, we can write:

\( x = \frac{8 \times 2520}{8 \times 7} \)

Cancel out the 8 from the numerator and the denominator:

\( x = \frac{2520}{7} \)

Now, divide 2520 by 7:

\( 2520 \div 7 \)

Let's perform the division:

• \( 25 \div 7 = 3 \) with a remainder of \( 25 - (7 \times 3) = 25 - 21 = 4 \).
• Bring down the next digit (2) to get 42.
• \( 42 \div 7 = 6 \).
• Bring down the last digit (0) to get 0.
• \( 0 \div 7 = 0 \).

So, \( 2520 \div 7 = 360 \).

Therefore, the value of x is 360.

\( x = 360 \)

The other number is 360.

Verification

Let's check if the HCF of 56 and 360 is 8 and the LCM is 2520.

HCF of 56 and 360:

Prime factorization of 56: \( 2^3 \times 7 \)

Prime factorization of 360: \( 36 \times 10 = (2^2 \times 3^2) \times (2 \times 5) = 2^3 \times 3^2 \times 5 \)

Common prime factors with the lowest power: \( 2^3 = 8 \). So, HCF(56, 360) = 8. This matches the given HCF.

LCM of 56 and 360:

Prime factorization of 56: \( 2^3 \times 7^1 \times 3^0 \times 5^0 \)

Prime factorization of 360: \( 2^3 \times 3^2 \times 5^1 \times 7^0 \)

Highest power of each prime factor: \( 2^3, 3^2, 5^1, 7^1 \)

LCM(56, 360) = \( 2^3 \times 3^2 \times 5^1 \times 7^1 = 8 \times 9 \times 5 \times 7 = 72 \times 35 \)

\( 72 \times 35 = 72 \times (30 + 5) = (72 \times 30) + (72 \times 5) = 2160 + 360 = 2520 \)

The LCM is 2520, which also matches the given LCM.

The product of the numbers is \( 56 \times 360 \). The product of HCF and LCM is \( 8 \times 2520 \).

\( 56 \times 360 = 20160 \)

\( 8 \times 2520 = 20160 \)

Since \( 56 \times 360 = 8 \times 2520 \), the relationship holds true, and our calculated other number, 360, is correct.

Conclusion

Using the relationship \( \text{HCF} \times \text{LCM} = \text{Product of the two numbers} \), we found that the other number is 360.

Revision Table: HCF and LCM Concepts

Additional Information: Finding HCF and LCM

Besides using the relationship with the product of numbers, HCF and LCM can be found using various methods:

• Prime Factorization Method:
  • Find the prime factorization of each number.
  • For HCF, multiply the common prime factors raised to the lowest power.
  • For LCM, multiply all prime factors (common and non-common) raised to the highest power.
• Division Method:
  • For HCF, use the Euclidean algorithm (repeated division).
  • For LCM, divide the numbers by common prime factors until no common factor exists, then multiply all divisors and the remaining numbers.

Understanding these methods is crucial for solving problems involving HCF and LCM.