Finding the Second Number Using LCM and HCF

This problem requires us to find one of two numbers when their Least Common Multiple (LCM), Highest Common Factor (HCF), and the other number are known. There is a fundamental relationship between the LCM and HCF of two positive integers and the product of the integers themselves.

Understanding the Relationship between LCM, HCF, and Numbers

For any two positive integers, say 'a' and 'b', the product of these two numbers is always equal to the product of their LCM and HCF. This can be written as:

Product of two numbers = Product of their LCM and HCF

Mathematically, this is expressed as:

\(a \times b = \text{LCM}(a, b) \times \text{HCF}(a, b)\)

Applying the Formula to the Given Problem

In this question, we are given:

• LCM of the two numbers = 70
• HCF of the two numbers = 7
• One number = 35
• Let the second number be \(x\).

Using the formula, we can set up the equation:

\(\text{One number} \times \text{Second number} = \text{LCM} \times \text{HCF}\)

\(35 \times x = 70 \times 7\)

Calculating the Second Number

Now, we need to solve this equation for \(x\):

First, calculate the product of LCM and HCF:

\(70 \times 7 = 490\)

So the equation becomes:

\(35 \times x = 490\)

To find \(x\), divide the product (490) by the known number (35):

\(x = \frac{490}{35}\)

Let's perform the division:

\(x = 14\)

Therefore, the second number is 14.

Verification

We can verify this by checking if the product of the two numbers (35 and 14) equals the product of the LCM (70) and HCF (7).

• Product of numbers: \(35 \times 14 = 490\)
• Product of LCM and HCF: \(70 \times 7 = 490\)

Since both products are equal (490), our calculation for the second number is correct.

The calculated second number is 14.

Revision Table: LCM and HCF Property

Additional Information: Finding LCM and HCF

While this problem uses the relationship, it's useful to know how to find LCM and HCF for two numbers. One common method is prime factorization.

• Prime Factorization Method:
  • Find the prime factorization of each number.
  • For HCF, take the product of the lowest powers of all common prime factors.
  • For LCM, take the product of the highest powers of all prime factors involved in either number.

Let's take the numbers 35 and 14 found in the problem:

• Prime factorization of 35: \(5^1 \times 7^1\)
• Prime factorization of 14: \(2^1 \times 7^1\)

Now find HCF and LCM:

• HCF(35, 14): The common prime factor is 7. The lowest power of 7 is \(7^1\). So, HCF = 7.
• LCM(35, 14): The prime factors involved are 2, 5, and 7. The highest power of 2 is \(2^1\), the highest power of 5 is \(5^1\), and the highest power of 7 is \(7^1\). So, LCM = \(2^1 \times 5^1 \times 7^1 = 2 \times 5 \times 7 = 70\).

These values (HCF=7, LCM=70) match the values given in the problem, further confirming that 14 is indeed the correct second number.