If the HCF of 408 and 1032 is expressible in the form 1032 × 2 408 P,find P?

Question

If the HCF of 408 and 1032 is expressible in the form 1032 × 2   408   P,find P?

Answer

Answer

Answer

Introduction:
To find the value of P in the expression 1032 × 2 + 408 + P, we need to determine the highest common factor (HCF) of 408 and 1032. The HCF is the largest number that evenly divides both numbers. Once we have the HCF, we can substitute it into the given expression to find the value of P.

Step 1: Prime Factorization:
To find the HCF, we start by performing prime factorization of both numbers. Prime factorization expresses a number as a product of its prime factors.

Prime factorization of 408:
408 = 2 × 2 × 2 × 3 × 17

Prime factorization of 1032:
1032 = 2 × 2 × 2 × 3 × 43

Step 2: Common Prime Factors:
Next, we identify the common prime factors between 408 and 1032. These common factors will contribute to the HCF.

Common prime factors: 2, 2, 2, 3

Step 3: Determine the HCF:
To find the HCF, we take the product of the common prime factors. In this case, it is 2 × 2 × 2 × 3 = 24.

Step 4: Substitute into the Expression:
Now that we have the value of the HCF, we substitute it into the given expression: 1032 × 2 + 408 + P.

Substituting the HCF (24) into the expression:

1032 × 2 + 408 + P = 1032 × 2 + 408 + 24

Simplifying the expression:

2064 + 408 + P = 2472 + P

Step 5: Solve for P:
To find the value of P, we subtract 2472 from both sides of the equation:

2064 + 408 + P - 2472 = 2472 + P - 2472

Simplifying the equation:

P + 408 - 408 = 0

P = 0

Conclusion:
After evaluating the expression and solving the equation, we find that P is equal to 0.

Answer

P= - 5

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