If 56 × 75 × 60 × 84 × 210 is expressed as 2^p × 3^q × 5^r × 7^s, then what is the value of [(p + q) / s] + r?

Introduction / Context:
Prime factorisation is a standard topic in number theory and quantitative aptitude. This question asks you to factor a product of several composite numbers into prime powers of 2, 3, 5 and 7, and then perform a simple arithmetic operation on the exponents. Such problems test your ability to break numbers into primes quickly and to manage exponents correctly during multiplication.

Given Data / Assumptions:

• The product is N = 56 × 75 × 60 × 84 × 210.
• N is expressed as 2^p × 3^q × 5^r × 7^s.
• We need to find the value of [(p + q) / s] + r.
• All prime factors involved are 2, 3, 5 and 7.

Concept / Approach:
The idea is to factor each number individually into primes and then add the exponents of like primes across the product. For example, if one factor contributes 2^3 and another contributes 2^2, then together they contribute 2^(3+2) = 2^5. After finding the total exponent for each prime, we substitute p, q, r and s into the given expression. Careful factorisation and accurate addition of exponents are essential to avoid mistakes.

Step-by-Step Solution:
Step 1: Factor each number into primes. 56 = 2^3 × 7. 75 = 3 × 5^2. 60 = 2^2 × 3 × 5. 84 = 2^2 × 3 × 7. 210 = 2 × 3 × 5 × 7. Step 2: Add exponents for each prime across all factors. For prime 2: total exponent p = 3 + 0 + 2 + 2 + 1 = 8. For prime 3: total exponent q = 0 + 1 + 1 + 1 + 1 = 4. For prime 5: total exponent r = 0 + 2 + 1 + 0 + 1 = 4. For prime 7: total exponent s = 1 + 0 + 0 + 1 + 1 = 3. Step 3: Evaluate the expression [(p + q) / s] + r = [(8 + 4) / 3] + 4. Step 4: Compute (8 + 4) / 3 = 12 / 3 = 4, then add r = 4 to get 4 + 4 = 8.

Verification / Alternative check:
You can verify the exponents by recombining them into a single number and checking a small part numerically. For example, check that the exponent of 2 is correct by counting the factors of 2 directly in each original number, or by dividing N by 2 repeatedly until it is no longer divisible by 2. Similar checks can be done for the other primes. All methods should yield p = 8, q = 4, r = 4 and s = 3, confirming that the computed expression equals 8.

Why Other Options Are Wrong:

• Option a (6) would correspond to incorrect exponent counts, perhaps missing some factors of 2 or 3.
• Option c (12) might come from adding p, q and r directly without dividing by s.
• Option d (10) could result from miscomputing (p + q) or s, leading to an incorrect intermediate value.
• Option e (14) is too large and suggests multiple arithmetic errors in handling exponents.

Common Pitfalls:
Students often misfactor one of the composite numbers, especially those like 84 or 210 that have several prime factors. Another common mistake is to forget that exponents add when multiplying powers with the same base. Some candidates rush through the arithmetic and miscalculate the final expression. To avoid such errors, always factor each number carefully, keep a running tally of exponents for each prime, and perform the final arithmetic step with attention.

Final Answer:
The value of [(p + q) / s] + r is 8.