If 313632 = p² × q⁵ × r⁴, where p, q and r are prime numbers, then what is the value of (p + q − 2r) ?

The correct answer is 7.

We are given the prime factorization equation:

$p^2\times q^5\times r^4=313632$

where p, q, and r are prime numbers.

Step 1: Perform the prime factorization of 313632.

Divide 313632 step-by-step by prime numbers:
313632 ÷ 2 = 156816
156816 ÷ 2 = 78408
78408 ÷ 2 = 39204
39204 ÷ 2 = 19602
19602 ÷ 2 = 9801
So, 313632 = 2⁵ × 9801

Now, factorize 9801:
Sum of digits of 9801 is 9 + 8 + 0 + 1 = 18, which is divisible by 9 (or 3).
9801 ÷ 3 = 3267
3267 ÷ 3 = 1089
1089 ÷ 3 = 363
363 ÷ 3 = 121
So, 9801 = 3⁴ × 121

Now, factorize 121:
121 = 11²

Combining all the prime factors together, we get:
313632 = 2⁵ × 3⁴ × 11²

Rearranging the factors to match the given form p² × q⁵ × r⁴:

${11}^2\times 2^5\times 3^4=p^2\times q^5\times r^4$

Step 2: Identify the values of p, q, and r.

Comparing powers on both sides:
p = 11 (since power is 2)
q = 2 (since power is 5)
r = 3 (since power is 4)

Step 3: Calculate the value of (p + q − 2r).

Substitute the values of p, q, and r into the expression:

$p+q-2r=11+2-2\left(3\right)$

$=13-6=7$

Thus, the value of (p + q − 2r) is 7.