"If x = 1/(2² × 3² × 4² × 5²), How Many Distinct Non-Zero Digits Does x Have?" — GMAT® Worked Solution
From Episode 46 of Real GMAT® Problems (The GMAT® Strategy Podcast). For the full strategy behind computation on the GMAT®, read: GMAT® Computation: When to Do the Math (and When to Let Go).
The Problem
Source: Official Guide for GMAT® Review, 11th Edition
If x = 122 × 32 × 42 × 52 is expressed as a decimal, how many distinct non-zero digits will x have?
(A) 1
(B) 2
(C) 3
(D) 7
(E) 10
Try it before reading on.
Why This Problem Is Hard
There's a lot going on. A large compound denominator. Exponents. The need to convert to a decimal. And a question about DISTINCT non-zero digits — which means we need to identify unique values, not just count all the digits.
There's no clean trick that gets you to the answer without some computation. This is one of those problems where, if you're going to get to 100% certainty, you're probably going to have to do a fair amount of math.
That's worth knowing upfront. On the GMAT®, recognizing when brute force is the right call — and having the patience to execute it — is itself a skill.
It's also worth naming the other option: if this type of problem is beyond your current traction, letting it go quickly and redirecting your time is a defensible choice. Missing a hard question costs less than rushing through familiar ones and getting those wrong.
Step 1: Find the Key Simplification
The denominator is 2² × 3² × 4² × 5².
We can multiply these in any order. Group the 2² and 5² together:
2² = 4 5² = 25 4 × 25 = 100
So we can rewrite the denominator as 100 × 3² × 4², and separate:
x = 1100 × 132 × 42
Why does this help? Multiplying or dividing by 100 (or any power of 10) just shifts the decimal point. It does not add any new non-zero digits.
So the question simplifies to: how many distinct non-zero digits appear in the decimal expansion of 132 × 42? The factor of 1100 won't change the answer.
Step 2: Compute the Remaining Denominator
3² = 9 4² = 16 9 × 16 = 144
We need to convert 1144 to a decimal.
Step 3: Long Division of 1 ÷ 144
We need 1 ÷ 144. Since 144 is larger than 1, 10, and 100, the first non-zero digit won't appear until the thousandths place.
Here's how that looks on paper:
0 . 0 0 6 9 4 4 ...
┌────────────────────────
144 │ 1 . 0 0 0 0 0 0
0 ← 144 > 1
0 ← 144 > 10
0 ← 144 > 100
─────
1 0 0 0 144 × 6 = 864
− 8 6 4 144 × 7 = 1008 ✗
─────
1 3 6 0 144 × 9 = 1296
− 1 2 9 6 144 × 10 = 1440 ✗
─────
6 4 0 144 × 4 = 576
− 5 7 6 144 × 5 = 720 ✗
─────
6 4 0 ← same remainder
− 5 7 6 → 4 repeats
─────
6 4 ...
The remainder 64 appeared twice in a row — which means the digit 4 will keep repeating.
Result: 1144 = 0.00694444...
Step 4: Count the Distinct Non-Zero Digits
The decimal is 0.0069444...
Non-zero digits: 6, 9, 4.
Are they distinct? Yes. Three unique values.
Account for the factor of 1/100 (from Step 1): that shifts the decimal point two more places — the digits don't change.
x = 0.0000694444...
The distinct non-zero digits are still 6, 9, and 4.
The answer is (C).
What to Take Away
2² × 5² = 100 is a move worth knowing. Any time you see a power of 2 and a power of 5 in a denominator, check whether they multiply to a power of 10. Powers of 10 only shift the decimal point — they don't add new digits. Spotting this early simplifies the problem significantly.
Sometimes you have to do the math. There's no shortcut that gets you from the denominator to the distinct digits without computing 1 ÷ 144. Patience matters here. Students who get this right are often the ones willing to set up the long division and work through it step by step.
Letting go is also correct. If you're in the exam, time is limited, and this problem doesn't have traction — it's okay to make an educated guess and move on. The time you spend struggling here might cost you credit on problems you could have gotten right. That's the trade worth thinking about.
Want the full strategy? Read: GMAT® Computation: When to Do the Math (and When to Let Go)
From Episode 46 of Real GMAT® Problems (The GMAT® Strategy Podcast).