Q 28. The HCF of two numbers 'x' and 'y' (x > y) is 7 and the sum is 56. How many distinct values of (x - y) are possible?

1

2

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8

3

Given the information:

1. The highest common factor (HCF) of two numbers \( x \) and \( y \) is 7.
2. Their sum is 56, so \( x + y = 56 \).
3. We are asked for the number of distinct values of \( x - y \) that are possible.

### Step-by-Step Solution

Since the HCF of \( x \) and \( y \) is 7, both \( x \) and \( y \) must be multiples of 7. Let:
\[
x = 7a \quad \text{and} \quad y = 7b
\]
where \( a \) and \( b \) are integers that are coprime (since 7 is their HCF).

Substituting \( x = 7a \) and \( y = 7b \) into \( x + y = 56 \), we get:
\[
7a + 7b = 56
\]
Dividing by 7:
\[
a + b = 8
\]

We now need to find possible integer values of \( a \) and \( b \) such that \( a + b = 8 \) and \( a > b \).

### Possible Pairs \((a, b)\) Satisfying \( a + b = 8 \)

The pairs \((a, b)\) with \( a > b \) are:
1. \( (7, 1) \)
2. \( (6, 2) \)
3. \( (5, 3) \)

### Calculating \( x - y \) for Each Pair

1. For \( (a, b) = (7, 1) \):
   \[
   x - y = 7a - 7b = 7 \times 7 - 7 \times 1 = 49 - 7 = 42
   \]

2. For \( (a, b) = (6, 2) \):
   \[
   x - y = 7a - 7b = 7 \times 6 - 7 \times 2 = 42 - 14 = 28
   \]

3. For \( (a, b) = (5, 3) \):
   \[
   x - y = 7a - 7b = 7 \times 5 - 7 \times 3 = 35 - 21 = 14
   \]

### Distinct Values of \( x - y \)

The distinct values of \( x - y \) are 42, 28, and 14. So, there are **3 distinct values** of \( x - y \) possible.

### Answer

The correct answer is:
\[
\boxed{3}
\]

Would you like further details or have any questions?

---

1. Why must \( x \) and \( y \) be multiples of their HCF?
2. Can the values of \( x \) and \( y \) be non-integers in this type of problem?
3. How can we confirm \( x > y \) based on the pairs \((a, b)\)?
4. Why is it necessary to ensure \( a \) and \( b \) are coprime?
5. What are other methods to find distinct values of \( x - y \)?

**Tip:** When working with HCF problems, always check if the numbers can be written as multiples of the HCF to simplify the problem.

The HCF of two numbers 'x' and 'y' (x > y) is 7 and the sum is 56. How many distinct values of (x - y) are possible?

1

2

4

8

3

Explore a solution to find the distinct values of (x - y) for two numbers with HCF 7 and sum 56. This problem covers number theory concepts, including HCF factorization and integer solutions for linear equations. Suitable for Grades 9-10.

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