Understanding the Problem of Number Pairs and HCF

The question asks us to find the number of unique pairs of positive integers that satisfy two conditions: their sum is 240, and their Highest Common Factor (HCF) is 15.

Let the two positive numbers be \(a\) and \(b\). We are given:

• \(a + b = 240\)
• HCF\((a, b) = 15\)

When the HCF of two numbers \(a\) and \(b\) is 15, it means that both \(a\) and \(b\) are multiples of 15. We can write \(a\) and \(b\) in terms of their HCF and two other numbers.

Let \(a = 15x\) and \(b = 15y\), where \(x\) and \(y\) are positive integers.

An important property when expressing numbers this way is that the numbers \(x\) and \(y\) must be coprime. Coprime means that their Highest Common Factor is 1, i.e., HCF\((x, y) = 1\) or gcd\((x, y) = 1\). This ensures that 15 is indeed the highest common factor of \(a\) and \(b\).

Setting up the Equation with HCF

Now substitute \(a = 15x\) and \(b = 15y\) into the sum equation:

\(15x + 15y = 240\)

We can factor out 15 from the left side:

\(15(x + y) = 240\)

To find the sum of \(x\) and \(y\), divide both sides by 15:

\(x + y = \frac{240}{15}\)

\(x + y = 16\)

Finding Coprime Pairs (x, y) that Sum to 16

We now need to find pairs of positive integers \((x, y)\) such that their sum is 16 and they are coprime (gcd\((x, y) = 1\)). Since \(x\) and \(y\) are interchangeable for the pair \(\{a, b\}\), we can list pairs \((x, y)\) where \(x \le y\) to avoid counting the same pair of numbers \(\{a, b\}\) twice, but we must ensure we check the coprime condition for each pair.

Let's list the pairs of positive integers \((x, y)\) whose sum is 16:

• (1, 15)
• (2, 14)
• (3, 13)
• (4, 12)
• (5, 11)
• (6, 10)
• (7, 9)
• (8, 8)

Checking the Coprime Condition (gcd(x, y) = 1)

Now we check the HCF (or gcd) for each pair \((x, y)\) to see which ones are coprime:

The pairs \((x, y)\) that are coprime are (1, 15), (3, 13), (5, 11), and (7, 9).

Each of these coprime pairs corresponds to a unique pair of positive numbers \((a, b)\) with HCF 15 and sum 240:

• For (x, y) = (1, 15): \(a = 15 \times 1 = 15\), \(b = 15 \times 15 = 225\). Pair: (15, 225). Sum = 15 + 225 = 240. HCF(15, 225) = 15.
• For (x, y) = (3, 13): \(a = 15 \times 3 = 45\), \(b = 15 \times 13 = 195\). Pair: (45, 195). Sum = 45 + 195 = 240. HCF(45, 195) = 15.
• For (x, y) = (5, 11): \(a = 15 \times 5 = 75\), \(b = 15 \times 11 = 165\). Pair: (75, 165). Sum = 75 + 165 = 240. HCF(75, 165) = 15.
• For (x, y) = (7, 9): \(a = 15 \times 7 = 105\), \(b = 15 \times 9 = 135\). Pair: (105, 135). Sum = 105 + 135 = 240. HCF(105, 135) = 15.

We found 4 such coprime pairs \((x, y)\). Each coprime pair corresponds to a distinct pair of numbers \(\{a, b\}\).

Therefore, there are 4 pairs of numbers satisfying the given conditions.

Conclusion on the Number of Pairs

Based on our analysis, there are 4 pairs of positive numbers whose sum is 240 and whose HCF is 15.

Revision Table: Key Concepts for Number Problems

Additional Information: General Approach for HCF and Sum/Product Problems

Problems involving the sum or product of two numbers and their HCF can often be solved using the approach demonstrated above.

General steps:

1. Represent the two numbers as \(a = \text{HCF} \times x\) and \(b = \text{HCF} \times y\), where \(x\) and \(y\) are positive integers and gcd\((x, y) = 1\).
2. Use the given condition (sum or product) to form an equation in terms of \(x\) and \(y\).
   • If sum \(S\) is given: \(\text{HCF}(x + y) = S \implies x + y = S / \text{HCF}\).
   • If product \(P\) is given: \((\text{HCF} \times x)(\text{HCF} \times y) = P \implies \text{HCF}^2 \times xy = P \implies xy = P / \text{HCF}^2\).
3. Find pairs of positive integers \((x, y)\) that satisfy the equation from step 2 AND the coprime condition gcd\((x, y) = 1\).
4. Each valid coprime pair \((x, y)\) corresponds to a unique pair of the original numbers \((a, b)\). Count the number of such coprime pairs.

This systematic approach helps ensure you find all possible pairs and correctly apply the properties of HCF and coprime numbers.