Challenging GMAT Quant Question 24

Numbers | Factorials & Prime Factorization

The given question is a challenging GMAT 700+ level quant problem solving question testing concepts in number properties. Concepts covered: Factorials and Prime Factorisation of numbers.

Question 24: If y is the highest power of a number 'x' that can divide 101! without leaving a remainder, then for which among the following values of x will y be the highest?

  1. 111
  2. 462
  3. 74
  4. 33
  5. 210

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Explanatory Answer | GMAT Numbers Practice Question

It is clear that all the answer options given are composite number. So, the divisor is not prime.

Approach to find the highest power of a composite number that divides n!

Step 1: Prime factorize the divisor, ‘x’ in this case.
Step 2: Compute the highest power of each of the prime factors that divides n!
Step 3: The highest power of x that divides n! is determined by the power of that prime factor which available in the least number.

Key Inference: The value of 'y' will be the highest for such an x whose highest prime factor is the smallest.

Prime Factorize numbers given in the 5 answer options

Option A: 111 = 3 × 37. The highest prime factor of 111 is 37.
Option B: 462 = 2 × 3 × 7 × 11. The highest prime factor of 462 is 11.
Option C: 74 = 2 × 37. The highest prime factor of 74 is 37.
Option D: 33 = 3 × 11. The highest prime factor of 33 is 11.
Option E: 210 = 2 × 3 × 5 × 7. The highest prime factor of 210 is 7.

210 is the number which has the smallest value of the highest prime factor among the 5 given options.
So, the value of y will be highest for 210.

Choice E is the correct answer.

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