If H.C.F. (56, 72) = 56x + 72y, then x and y are equal to :

Q: If H.C.F. (56, 72) = 56x + 72y, then x and y are equal to :

To determine the values of $ x $ and $ y $ in the equation $ \text{H.C.F.}(56, 72) = 56x + 72y $, we need to understand the concept of expressing the highest common factor (H.C.F.) as a linear combination of the two numbers. This is based on the principle of the Euclidean algorithm, which states that the H.C.F. of two numbers can be expressed as a linear combination of those numbers. 1. **Calculation of H.C.F.:** - The H.C.F. of 56 and 72 can be found using the Euclidean algorithm: - $ 72 \div 56 = 1 $ remainder $ 16 $ - $ 56 \div 16 = 3 $ remainder $ 8 $ - $ 16 \div 8 = 2 $ remainder $ 0 $ - Therefore, the H.C.F. is 8. 2. **Expressing H.C.F. as a Linear Combination:** - We need to express 8 as a linear combination of 56 and 72, i.e., $ 8 = 56x + 72y $. 3. **Verification of Option 3:** - Let's check if $ x = 4 $ and $ y = -3 $ satisfy the equation: - Substitute $ x = 4 $ and $ y = -3 $ into the equation: \[ 56(4) + 72(-3) = 224 - 216 = 8 \] - The equation holds true, confirming that $ x = 4 $ and $ y = -3 $ is a valid solution. 4. **Explanation of Incorrect Options:** - **Option 1: $ x = 4, y = 3 $** - Substituting these values gives: \[ 56(4) + 72(3) = 224 + 216 = 440 \] - This does not equal 8, so this option is incorrect. - **Option 2: $ x = -4, y = 3 $** - Substituting these values gives: \[ 56(-4) + 72(3) = -224 + 216 = -8 \] - This does not equal 8, so this option is incorrect. - **Option 4: $ x = -4, y = -3 $** - Substituting these values gives: \[ 56(-4) + 72(-3) = -224 - 216 = -440 \] - This does not equal 8, so this option is incorrect. In conclusion, Option 3 is the correct answer because it satisfies the equation $ 56x + 72y = 8 $ with $ x = 4 $ and $ y = -3 $.

Question

Accepted Answer

To determine the values of $ x $ and $ y $ in the equation $ \text{H.C.F.}(56, 72) = 56x + 72y $, we need to understand the concept of expressing the highest common factor (H.C.F.) as a linear combination of the two numbers. This is based on the principle of the Euclidean algorithm, which states that the H.C.F. of two numbers can be expressed as a linear combination of those numbers.

1. **Calculation of H.C.F.:**
   - The H.C.F. of 56 and 72 can be found using the Euclidean algorithm:
     - $ 72 \div 56 = 1 $ remainder $ 16 $
     - $ 56 \div 16 = 3 $ remainder $ 8 $
     - $ 16 \div 8 = 2 $ remainder $ 0 $
   - Therefore, the H.C.F. is 8.

2. **Expressing H.C.F. as a Linear Combination:**
   - We need to express 8 as a linear combination of 56 and 72, i.e., $ 8 = 56x + 72y $.

3. **Verification of Option 3:**
   - Let's check if $ x = 4 $ and $ y = -3 $ satisfy the equation:
     - Substitute $ x = 4 $ and $ y = -3 $ into the equation:
     $$
     56(4) + 72(-3) = 224 - 216 = 8
     $$
   - The equation holds true, confirming that $ x = 4 $ and $ y = -3 $ is a valid solution.

4. **Explanation of Incorrect Options:**
   - **Option 1: $ x = 4, y = 3 $**
     - Substituting these values gives:
     $$
     56(4) + 72(3) = 224 + 216 = 440
     $$
     - This does not equal 8, so this option is incorrect.
   
   - **Option 2: $ x = -4, y = 3 $**
     - Substituting these values gives:
     $$
     56(-4) + 72(3) = -224 + 216 = -8
     $$
     - This does not equal 8, so this option is incorrect.
   
   - **Option 4: $ x = -4, y = -3 $**
     - Substituting these values gives:
     $$
     56(-4) + 72(-3) = -224 - 216 = -440
     $$
     - This does not equal 8, so this option is incorrect.

In conclusion, Option 3 is the correct answer because it satisfies the equation $ 56x + 72y = 8 $ with $ x = 4 $ and $ y = -3 $.