Understanding HCF, LCM, and Number Relationships

The problem involves finding the smaller of two numbers given their Highest Common Factor (HCF) and information about their Least Common Multiple (LCM). We need to use the fundamental relationship between two numbers, their HCF, and their LCM.

Key Relationship Between HCF and LCM

For any two positive integers, say 'a' and 'b', the product of the numbers is equal to the product of their HCF and LCM.

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

Another important concept is that if the HCF of two numbers 'a' and 'b' is \(h\), then the numbers can be expressed as \(a = hx\) and \(b = hy\), where \(x\) and \(y\) are integers that are coprime (their HCF is 1). In this case, their LCM is given by \( \text{LCM}(a, b) = hxy \).

Analyzing the Given Information

We are given:

• HCF of the two numbers is 17. So, \(h = 17\).
• The other two factors of their LCM are 11 and 19.

The LCM of two numbers is the product of their HCF and the other coprime factors from each number. The statement "other two factors of their LCM are 11 and 19" implies that the LCM can be written as HCF \( \times \) 11 \( \times \) 19. This is because 11 and 19 are given as factors of the LCM besides the HCF, and they must represent the \(x\) and \(y\) parts from the formula \( \text{LCM} = hxy \).

Since 11 and 19 are prime numbers, they are coprime. Therefore, we can consider these as the coprime factors \(x\) and \(y\).

• Let \(x = 11\)
• Let \(y = 19\)

Using the representation of the numbers \(a = hx\) and \(b = hy\), we can find the two numbers.

Calculating the Two Numbers

The two numbers are:

• First number \(a = hx = 17 \times 11\)
• Second number \(b = hy = 17 \times 19\)

Let's calculate their values:

• \(a = 17 \times 11\)

\(17 \times 11 = 110 + 77 = 187\)

• \(b = 17 \times 19\)

\(17 \times 19 = 190 + 133 = 323\)

So the two numbers are 187 and 323.

Identifying the Smaller Number

Comparing the two numbers, 187 and 323, the smaller number is 187.

Verification

Let's quickly verify our answer:

• Numbers are 187 and 323.
• HCF(187, 323): \(187 = 17 \times 11\), \(323 = 17 \times 19\). The common factor is 17. HCF = 17. (Correct)
• LCM(187, 323): The LCM is \(17 \times 11 \times 19\). The HCF is 17. The other two factors in the LCM are 11 and 19. (Correct)

The calculations match the given information, and the smaller number is 187.

Revision Table: HCF and LCM Concepts

Additional Information: Prime Factorization Method

The prime factorization method is often used to find the HCF and LCM of numbers. Let's consider our numbers 187 and 323.

• Prime factorization of 187: \(187 = 11 \times 17\)
• Prime factorization of 323: \(323 = 17 \times 19\)

To find the HCF, we look for common prime factors and take the lowest power:

• Common prime factor is 17.
• HCF(187, 323) = \(17^1 = 17\).

To find the LCM, we take all prime factors from both numbers and use the highest power:

• Prime factors are 11, 17, and 19.
• LCM(187, 323) = \(11^1 \times 17^1 \times 19^1 = 11 \times 17 \times 19 = 187 \times 19 = 3553\).

Notice that the LCM (\(17 \times 11 \times 19\)) consists of the HCF (17) and the 'other' factors (11 and 19), as stated in the problem. This confirms our approach was correct.