Finding Pairs of Numbers with Given Sum and HCF

The problem asks us to find the number of unique pairs of positive integers whose sum is 1215 and whose Highest Common Factor (HCF) is 81.

Understanding the Relationship between Numbers and their HCF

If the HCF of two numbers is 81, it means both numbers are multiples of 81. Let the two numbers be \(a\) and \(b\). We can express them as:

• \(a = 81x\)
• \(b = 81y\)

where \(x\) and \(y\) are positive integers. A crucial property here is that for 81 to be the *highest* common factor of \(a\) and \(b\), the factors \(x\) and \(y\) must be coprime. This means their HCF must be 1 (\(\text{HCF}(x, y) = 1\)).

Using the Sum Condition

We are given that the sum of the two numbers is 1215. So, we have:

\(a + b = 1215\)

Substitute the expressions for \(a\) and \(b\):

\(81x + 81y = 1215\)

Factor out the common factor 81:

\(81(x + y) = 1215\)

Now, we can find the value of \((x + y)\) by dividing the sum by the HCF:

\(x + y = \frac{1215}{81}\)

Calculating the Sum of x and y

Let's perform the division:

\(1215 \div 81\)

We can simplify this by dividing both numbers by common factors, like 9:

• \(1215 \div 9 = 135\)
• \(81 \div 9 = 9\)

So, the expression becomes \(\frac{135}{9}\). Now, divide 135 by 9:

\(135 \div 9 = 15\)

Thus, we have:

\(x + y = 15\)

Finding Coprime Pairs (x, y) that Sum to 15

We need to find pairs of positive integers \((x, y)\) such that their sum is 15 and their HCF is 1. To avoid counting the same pair of numbers \((a, b)\) twice (e.g., \(a=81x, b=81y\) is the same pair as \(a=81y, b=81x\)), we can assume \(x \le y\) when listing the pairs of \((x, y)\).

Let's list the pairs of positive integers \((x, y)\) such that \(x + y = 15\) and check if they are coprime:

The pairs \((x, y)\) that satisfy both conditions (\(x + y = 15\) and \(\text{HCF}(x, y) = 1\)) are (1, 14), (2, 13), (4, 11), and (7, 8).

Counting the Valid Pairs

Each valid coprime pair \((x, y)\) corresponds to a unique pair of numbers \((81x, 81y)\) whose sum is 1215 and HCF is 81. The valid pairs are:

• \((x, y) = (1, 14) \implies\) Numbers are \((81 \times 1, 81 \times 14) = (81, 1134)\). Sum = 1215, HCF = 81.
• \((x, y) = (2, 13) \implies\) Numbers are \((81 \times 2, 81 \times 13) = (162, 1053)\). Sum = 1215, HCF = 81.
• \((x, y) = (4, 11) \implies\) Numbers are \((81 \times 4, 81 \times 11) = (324, 891)\). Sum = 1215, HCF = 81.
• \((x, y) = (7, 8) \implies\) Numbers are \((81 \times 7, 81 \times 8) = (567, 648)\). Sum = 1215, HCF = 81.

There are 4 such pairs of numbers.

Conclusion

By using the property that two numbers with a given HCF can be expressed as the HCF multiplied by two coprime integers, we were able to determine the sum of these coprime integers and systematically find the pairs that satisfy the condition. There are 4 such pairs.

Revision Table - Sum and HCF Problems

Additional Information - HCF and Coprime Numbers

Highest Common Factor (HCF): The HCF (also known as the Greatest Common Divisor or GCD) of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder. For example, the HCF of 12 and 18 is 6.

Coprime Numbers: Two integers are said to be coprime (or relatively prime) if their only positive common divisor is 1. This means their HCF is 1. For example, 7 and 10 are coprime because \(\text{HCF}(7, 10) = 1\). Prime numbers are not necessarily coprime to each other (e.g., HCF(3, 6)=3), but any two distinct prime numbers are always coprime.

The property used in the solution, relating numbers with a specific HCF to coprime factors, is fundamental in solving problems involving HCF and relationships between numbers like their sum or product.