How to Find the Highest Power of a Divisor in a Factorial on GMAT?

Short Answer

To find the highest power of a divisor in n!, break the divisor into primes. Count occurrences of each prime using division, then take the limiting prime. For example, in 40!, 6 = 2 × 3. Counting 3s gives 18, so 6¹⁸ divides 40!.

Overview

In your GMAT prep, learning to work with factorials opens up fascinating insights into number patterns. One elegant application is finding the highest power of a divisor within a factorial. Instead of focusing on the entire base, you look at its prime factors and count how often they appear. With a little practice, this method becomes simple, quick, and reliable. Regular GMAT practice helps you internalize the approach, so what first seems abstract soon feels natural, giving you both clarity and confidence on test day.

 

Understanding How to Find the Highest Power of a Divisor in a Factorial

One of the most useful concepts in factorial-based questions on the GMAT is determining the highest power of a given divisor in n!. At first, many students guess small answers because they look at the divisor as a whole. For example, when asked for the highest power of 6 in 40!, answers like 2 or 6 may seem reasonable. However, factorials generate many layers of prime factors, and the correct answer is much larger. The key lies in breaking the divisor into its prime components and then counting how often those primes appear within the factorial.

Why Prime Factorization Matters

The number 6 is not prime. It is 2 × 3. The task, therefore, is not about counting 6s directly but about identifying which of these primes controls the maximum power. The number of 2s in 40! will be abundant, but 3s will be fewer. The highest power of 6 in 40! is determined entirely by the number of 3s present.

Counting the Occurrences

Question: If 3^x is a factor of 40!, what is the highest possible integer value of x?

To find how many times 3 occurs in 40!, we use the method of division.

• Divide 40 by 3, which gives 13. That means there are 13 multiples of 3 up to 40.
• Next, divide 40 by 9. This gives 4, which accounts for the extra 3s in numbers like 9, 18, and 27.
• Then, divide 40 by 27. This gives 1, for the triple 3 contained in 27.

Add them together: 13 + 4 + 1 = 18.

The answer is 18. So, 6 raised to the power 18 divides 40! completely.

Illustration with Another Example

Question: What highest integer power of 15 perfectly divides 100!?

The problem requires us to find the highest power of 15 in 100!.

Here, 15 = 3 × 5.

Since 5 is the higher prime, it will appear less frequently and therefore controls the answer.

Counting 5s in 100!:

• 100 ÷ 5 = 20
• 100 ÷ 25 = 4

Adding these gives 24.

Thus, the highest power of 15 in 100! is 24.

Practicing the Method

Once you understand the rule, practice becomes simple. Here are a few illustrations:

• Highest power of 7 in 50!: 50 ÷ 7 = 7, then 7 ÷ 7 = 1. Total = 8.
• Highest power of 5 in 50!: 50 ÷ 5 = 10, then 10 ÷ 5 = 2. Total = 12.
• Highest power of 11 in 150!: 150 ÷ 11 = 13, then 13 ÷ 11 = 1. Total = 14.

Composite Divisors and Special Cases

Sometimes the divisor itself is composite. For example, 70 = 2 × 5 × 7. The deciding factor is again the highest prime, which is 7. Counting 7s in 50! gives the result of 8. Interestingly, this is the same as the highest power of 7 alone.

Another case is when the divisor involves a prime raised to a power, such as 9 = 3².

Counting all 3s in 50! gives 22, but since each 9 requires two 3s, we divide 22 by 2. The answer is 11.

Similarly, for 16 = 2⁴, after counting all 2s in 50! (which gives 47).

We divide 47 by 4. The answer is 11.

Deciding Between Multiple Primes

Occasionally, the divisor has two primes, and it is unclear which controls the answer. For example, in 63 = 3² × 7, both pairs of 3s and individual 7s must be considered. In such cases, always calculate both possibilities and choose the smaller value.

Example:

For 50!, pairs of 3s yield 11, while 7s yield 8.

The smaller value, 8, is the correct answer.

This principle ensures accuracy when confusion arises.

Parting Thought

The exercise of finding the highest power of a divisor in a factorial is not just a mathematical trick, it is a way of training the mind to see structure where others see confusion. The GMAT rewards those who can slow down, break complexity into parts, and trust logic over instinct. This habit of disciplined reasoning becomes invaluable not only on test day but also later, in the stimulating MBA admission process, where clarity, patience, and careful judgment are equally essential. Numbers may test your skills, but it is perspective that defines your success.

 

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