The perils of non-essential formulas – a case study

By Brent Hanneson - May 12, 2022

In my last article, I recommended that students avoid memorizing a lot of non-essential (i.e., non-core) formulas for several reasons:  

• Many provide no value at all. 
• Many are too specific and don’t prepare you for alterations in a question.  
• Learning and practicing non-essential formulas increases your prep time and prevents you from mastering the core tools the test-makers expect you to master. 
• The test-makers are unlikely to create new questions that reward students for memorizing non-essential formulas

Let’s see how these points apply to the following question from the Official Guide:

If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?

(A) 10

(B) 12

(C) 14

(D) 16

(E) 18

This question made its first public appearance well before 2010, and shortly after it showed up on the GMAT forums, someone introduced a formula that would make questions exactly like this super easy to solve (once students have practiced it sufficiently).

The formula goes something like this:

If p is a prime number, then the maximum power of p to divide into n! equals n/p + n/p² + n/p³ + . . . . (continue the process until p^x > n)

Note: For each division, we record the quotient and ignore the remainder. 

Applying the formula to the above question, we get:

30/3 = 10 (with no remainder) 

30/3² = 30/9 = 3 (with remainder 3, which we ignore) 

30/3³ = 30/27 = 1 (with remainder 3, which we ignore) 

30/3⁴ = 30/81. Stop here, because 3⁴ > 30.

So, the maximum power of 3 that divides into 30! = 10 + 3 + 1 = 14 

Answer: C 

Since the official question was made public, dozens of similar (but unofficial) questions have popped up on the GMAT forums, where students familiar with the formula are delighted to apply this seemingly-useful tool. 

Here’s the first problem: 

Now that the formula is well established in the GMAT community, it’s doubtful the test-makers will create more questions of this nature (i.e., determining the greatest power of a prime to divide into a factorial), because the GMAT quant section is specifically designed to measure your quantitative reasoning skills, not your formula-memorizing skills. 

Don’t forget the test-makers’ statement that appears in every GMAT Official Guide: 

Knowledge of basic math, while necessary, is seldom sufficient for answering GMAT questions. Unlike traditional math problems you may have encountered in school, GMAT Quantitative Reasoning questions require you to apply your knowledge of math. For example, rather than asking you to demonstrate your knowledge of prime factorization by listing a number’s prime factors, a GMAT question may require you to apply your knowledge of prime factorization and exponents to simplify an algebraic expression with a radical.

For every official quant question, the test-makers expect us to apply quantitative reasoning and/or one or more of the core tools listed in the Official Guide. 

SPOILER: The greatest-power-of-a-prime-to-divide-some-factorial formula isn’t among those core tools.

In order to solve the original question, we must understand how divisibility is related to prime factorization (one of the core tools in the Official Guide). In fact, that one easy-to-understand property/tool can be used to solve the original question as well as a wide variety of other official questions, including this, this, this, this, this, this, and this (to name just a few). 

So, if we can solve so many different questions by applying just one of the GMAT’s core tools, why waste time memorizing and practicing super-specific non-core formulas? 

Aside: It’s worth noting that the “greatest-power-of-a-prime-to-divide-some-factorial” formula can be used to help find the prime factorization of a factorial, but the technique is only marginally faster than the conventional method. More importantly, if students don’t understand how/why the formula works, and they don’t understand the implications of the formula, they won’t know how to modify their solution if the details of a question aren’t perfectly aligned with the formula. 

For example, how would you apply the formula to the original question if it were altered such that p = product of EVEN integers from 2 to 30 inclusive? Similarly, what would you do if p = product of the integers from 12 to 37 inclusive? 

Since the conventional prime factorization technique can easily handle these alterations (and many others), the non-core formula provides no value.

Okay, I memorized an unnecessary formula. What’s the big deal? 

The official core tools are ridiculously versatile. So, if you decide to add any non-essential formulas to your mathematical toolbox, those formulas should provide significant value beyond what the core tools offer. If they don’t, then all you’re doing is extending your prep time. (Reminder: In a 2020 survey, the GMAT test-makers found that the median prep time for 700+ scorers was 100 hours).

For example, if you insist on memorizing the “biggest-power-of-a-prime-to-divide-some-factorial” formula, then you’ll also need to practice how and when to apply this unnecessary formula, when all you needed to do was learn how to apply the core concept of prime factorization to divisibility questions.   

But here’s the biggest reason to avoid memorizing a bunch of non-core formulas:

Unnecessary formulas serve as crutches that can actually make you less effective at solving GMAT math questions. 

Consider these two questions from the 2021 edition of the Official Guide for Quantitative Review: 

Question #204: What is the greatest positive integer n such that 5ⁿ divides 10! – (2)(5!)²? 

(A) 2

(B) 3

(C) 4

(D) 5

(E) 6

Question #207: If n = 9! – 6⁴, which of the following is the greatest integer k such that 3^k is a factor of n?

(A) 1

(B) 3

(C) 4

(D) 6

(E) 8

Both questions ask us to find the maximum power of a prime to divide into an expression involving factorials, but due to some minor alterations, our hard-to-memorize formula no longer applies (at least not directly). 

There are two potential headaches for students who memorized the formula only to be presented with these slightly modified questions on test day (note: both questions were once on the real exam): 

Headache #1: Since the questions closely resemble the original question the formula was based on, some students would have tried (unsuccessfully) to apply the formula. 

Headache #2: Students who recognized there was no direct way to apply the formula were forced to either modify the formula or find another approach. Had they resisted the temptation to add this formula to their mathematical toolbox, they might have had more time to master the core tools the test-makers expect us to master. 

Core tools to the rescue

As you probably guessed, the GMAT’s core tool of applying prime factorization to divisibility questions (which we used to solve the original question and others) can also be applied to questions #204 and #207 above. 

It’s all about APPLYING concepts

I should mention that there is a way to partially apply the greatest-power-of-a-prime-to-divide-some-factorial formula to help solve the two new questions above.  

Can you see how? If you’re going to keep that formula in your mathematical toolbox, then you should figure that out before test day. 

Last words 

Memorizing a lot of non-essential formulas significantly increases your prep time, and you’ll find that many of those formulas are so specific that it’s difficult to modify them to handle questions that don’t align perfectly with the setup the formula requires. Conversely, the official core tools are flexible enough to handle every official question. 

If you learn how to do more with less, you’ll make your prep easier, faster and more effective. 

 I want to thank Ian Stewart, GMAT mathematician extraordinaire, for reviewing this article (any remaining errors are my own).