Definition
Digit factorials
Sum of d! for each digit equals n.
Check Strong Number in Python
Overview
What You’ll Learn
A strong number (digital factorial) equals the sum of the factorials of its digits. Classic examples: 1, 2, 145, 40585. Non-examples: 10 (1!+0!=2), 99 (huge factorial sum). This tutorial covers a 0..9 factorial lookup, a live check, worked Python examples, edge cases, and complexity.
Lookup 0..9
Precompute
Avoid recomputing factorial each time.
145 Classic
1!+4!+5!
1 + 24 + 120 = 145.
1 and 2
Also strong
1! = 1 and 2! = 2.
Live Preview
Try 145 / 10
See digit factorial terms.
Not Armstrong
Powers vs !
Different digit tricks.
Introduction
A strong number equals the sum of the factorials of its digits. So 145 = 1! + 4! + 5! = 1 + 24 + 120, while 10 fails because 1! + 0! = 2.
Interviews love a small lookup table for 0! through 9!, then a digit loop with % 10 and // 10. That is fast, clear, and easy to dry-run on a whiteboard.
Why it matters?
It combines digit extraction with factorial basics — and shows why precomputing beats recomputing.
Key Highlights
Sum of d!
Equals the number.
Lookup Table
0! … 9! once.
1, 2, 145
In range 1..200.
vs Armstrong
Factorials, not powers.
In short: precompute fact[0..9], sum fact[digit] for every digit, and compare with n.
Foundation
📝 Problem & Approach
Given a positive integer n, decide whether the sum of factorials of its digits equals n.
python
# 145 -> 1! + 4! + 5! = 1 + 24 + 120 = 145 strong
# 2 -> 2! = 2 strong
# 10 -> 1! + 0! = 2 not strong
# 99 -> 9! + 9! = 725760 not strongInputs & Outputs
| Item | Type | Description |
|---|---|---|
n | int | Value to test (n >= 1 in this tutorial). |
| Return | bool | True when sum of digit factorials equals n. |
fact | list | Lookup for 0! through 9!. |
Minimal workflow
Pseudocode
fact = [1,1,2,6,24,120,720,5040,40320,362880]
function isStrong(n):
sum = 0
x = n
while x > 0:
d = x mod 10
sum = sum + fact[d]
x = floor(x / 10)
return sum == nMethod comparison
| Method | Idea | Notes |
|---|---|---|
| Lookup + digit loop | Precompute 0..9, sum fact[d] | Interview default |
| Range scan | Call is_strong on each i | Lists 1 2 145 in 1..200 |
| Trace terms | Print each d! contribution | Great for debugging |
Cheat Sheet
⚡ Quick Reference
| Goal | Pattern |
|---|---|
| Lookup | fact = [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880] |
| Next digit | digit = n % 10 |
| Add factorial | total += fact[digit] |
| Drop digit | n //= 10 |
| Verdict | return total == original |
| Early stop | if total > original: return False |
Compare
📋 Check vs Range vs Trace
Same definition — different packaging.
Single check
is_strong(145)Lookup + digit loop
Range
1..200Finds 1 2 145
Trace
print d!Shows each term
vs Armstrong
! vs ^Factorials, not powers
Context
When This Problem Shows Up
Reach for a strong check when digit factorials meet equality.
Interview warm-ups
Definition + lookup + digit loop.
Range listing
Find strong values in a band.
Factorial practice
Pairs with factorial tutorials.
Contrast Armstrong
Same digit loop, different op.
Not for 0
Most beginner defs start at n >= 1.
Key benefit: one memorable formula — sum of digit factorials — with a tiny constant-size lookup.
Interactive
🔮 Live Preview
Sums digit factorials with a 0..9 lookup and reports the strong verdict.
Hands-On
Examples Gallery
Three complete Python programs — check 145, list strong numbers from 1 to 200, and print digit-factorial traces for candidates. Click View Output to reveal sample console results.
📚 Getting Started
A lookup table plus a digit loop is the interview-friendly approach.
Example 1 — Check a Single Number
Precompute 0!..9!, walk digits, and compare the factorial sum with the original value.
python
def is_strong_number(n: int) -> bool:
fact = [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
original = n
total = 0
while n > 0:
digit = n % 10
total += fact[digit]
n //= 10
return total == original
number = 145
if is_strong_number(number):
print(f"{number} is a Strong Number.")
else:
print(f"{number} is not a Strong Number.")How It Works
Digits of 145 are 1, 4, and 5. Factorials are 1, 24, and 120, which sum to 145.
⚡ Hunting in a Range
Reuse the helper to list nearby strong values.
Example 2 — Strong Numbers from 1 to 200
Scan the band and print matches. Within 1..200 you only get 1, 2, and 145.
python
def is_strong_number(n: int) -> bool:
fact = [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
original = n
total = 0
while n > 0:
digit = n % 10
total += fact[digit]
n //= 10
return total == original
print("Strong Numbers in the Range 1 to 200:")
for i in range(1, 201):
if is_strong_number(i):
print(i, end=" ")
print()How It Works
1 and 2 are trivial strong numbers; 145 is the first multi-digit hit. The next famous one, 40585, sits well above 200.
Example 3 — Trace Digit Factorials for Candidates
Print each digit’s factorial contribution so you can see why a value is strong or not.
python
FACT = [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
def factorial_sum_terms(n: int):
terms = []
total = 0
x = n
while x > 0:
d = x % 10
total += FACT[d]
terms.append(f"{d}!={FACT[d]}")
x //= 10
terms.reverse()
return total, terms
for n in [2, 10, 145, 99]:
total, terms = factorial_sum_terms(n)
label = "strong" if total == n else "not strong"
print(f"{n}: {' + '.join(terms)} = {total} -> {label}")How It Works
10 fails because 0! is 1, not 0. 99 blows past the original value immediately because 9! is already huge.
🧠 How the Algorithm Decides
1
Build fact[0..9]
Precompute once; digits never need more.
Lookup
2
Extract digits
Use % 10 and // 10.
Loop
3
Add fact[digit]
Accumulate the factorial sum.
Sum
=
Compare with n
Equal means strong; otherwise not.
Trace
🔎 Worked Walkthrough — 145 vs 10
Compare a classic yes case with a common no case involving 0!.
| n | Digits | Factorial sum | Verdict |
|---|---|---|---|
145 | 1, 4, 5 | 1 + 24 + 120 = 145 | Strong |
2 | 2 | 2 = 2 | Strong |
10 | 1, 0 | 1 + 1 = 2 | Not strong |
99 | 9, 9 | 362880 + 362880 | Not strong |
Remember: 0! = 1, which trips people who expect zero.
Use Cases
Where strong checks show up beyond the interview prompt.
1. Interview Classics
Digit factorials equality.
Example: is_strong(145).
2. Range Listing
Find strong values in a band.
Example: 1 2 145.
3. Factorial Warm-up
Pairs with factorial tutorials.
Example: related links.
4. Contrast Armstrong
Same digits, different ops.
Example: FAQ.
5. Debugging Traces
Print each d! term.
Example: Example 3.
6. Next: Condense
Continue the interview chain.
Example: related CTA.
Pro Tip: open with “n equals the sum of factorials of its digits” and write the 0..9 table first.
Advantages
Why the lookup-table approach works well for beginners and interviews.
1. Tiny Constant Table
Only ten factorials ever matter.
2. Easy to Trace
Dry-run 145 on paper in seconds.
3. Fast Digit Loop
O(digits) with O(1) extras.
4. Early Exit Option
Stop if the running sum exceeds n.
Pro Tip: mention early-stop as an optional optimization after the clear baseline loop.
Usage Tips
Small habits that keep strong-number solutions interview-ready.
1. Save Original n
You destroy n while extracting digits.
2. Precompute Once
Never recompute factorial per digit.
3. Remember 0! = 1
It is why 10 is not strong.
4. Know 1..200 Hits
Expect 1, 2, and 145.
5. Contrast Armstrong
Say the difference out loud in interviews.
Pro Tip: sanity-check 1, 2, 10, 145, and 99 — if those five behave, your logic is solid.
Common Pitfalls
Mistakes that commonly break strong-number programs.
1. Treating 0! as 0
0! is 1 by definition.
→ Put 1 at fact[0].
2. Recomputing Factorial
Nested factorial loops per digit.
→ Use a lookup list.
3. Comparing Against Destroyed n
Forgetting to save original.
→ Keep original = n.
4. Confusing with Armstrong
Using powers instead of factorials.
→ Say the difference explicitly.
5. Calling 0 Strong
Outside this tutorial’s n >= 1 focus.
→ Follow the problem statement.
Edge Cases
Handle these before claiming the check is complete.
n = 0
Usually excluded
Most interview versions start from n >= 1.
n = 1, 2
Strong
1! = 1 and 2! = 2.
n = 10
Not strong
1! + 0! = 2.
Performance
Use a lookup
Do not recompute factorial often.
145
Classic yes
1! + 4! + 5! = 145.
40585
Larger famous case
Beyond the 1..200 list.
Theory
⚖️ Facts Worth Knowing
Handy follow-ups interviewers sometimes ask.
- Also called digital factorial numbers. Same idea, different name.
- Known base-10 examples. 1, 2, 145, and 40585.
- 0! = 1. Critical for any number containing digit 0.
- Not Armstrong. Armstrong uses digit powers; strong uses factorials.
Drill
🎯 Practice Problems
Try these variations to lock in the pattern.
1. Prove 145
- 1! + 4! + 5!
- Show = 145
2. Reject 10
- Use 0! = 1
- Confirm sum = 2
3. List 1..200
- Reproduce Example 2
- Expect 1 2 145
4. Trace 40585
- Optional stretch
- Verify digit factorials
Important
Notes
- Definition: strong means the sum of digit factorials equals n.
- Lookup: precompute factorials for 0..9 once.
- Range check: in 1..200 you should get
1 2 145. - Optimization: early-stop if the running sum exceeds the original number. Still prefer the clear baseline first.
Quick Takeaway: n is strong when sum(fact[digit] for each digit) == n.
Analysis
⏱️ Time and Space Complexity
| Task | Time | Extra space |
|---|---|---|
| Check one n | O(d) (d = digits) | O(1) |
| Scan 1..U | O(U log U) | O(1) |
| Lookup table | build once | 10 integers |
Digit count grows like log10 n, so a single check is essentially linear in the number of digits.
🎉 Conclusion
A strong number equals the sum of the factorials of its digits. Precompute 0! through 9!, walk the digits, and compare — remembering that 0! = 1.
Practice the three examples above, then continue to condensing a number.
Sum of digit factorials equals n.
Pro Tips
💡 Best Practices
✅ Do
- Precompute 0!..9!
- Save original n
- Treat 0! as 1
- Dry-run 145
- Know 1 2 145 in 1..200
❌ Don’t
- Recompute factorial every digit
- Confuse with Armstrong
- Assume 0! = 0
- Compare after destroying n
- Ignore the problem’s n >= 1 rule
Summary
Key Takeaways
Knowledge Unlocked
Five things to remember about strong numbers
Classify numbers whose digit factorials sum to themselves.
5
Core concepts 01
Rule
sum of digit !
Definition 02
Table
0!..9! lookup
Method 03
Trap
0! = 1
Edge 04
List
1 2 145
Check 05
Cost
O(digits)
Analysis❓ Frequently Asked Questions
A strong number equals the sum of the factorials of its digits. Example: 145 = 1! + 4! + 5!.
Yes. 1! = 1 and 2! = 2.
Usually no for beginner definitions focused on positive integers, because 0! = 1.
Digits are only 0 to 9, so precomputing 0! to 9! avoids repeated factorial calculations.
1, 2, and 145.
No. Armstrong numbers use powers of digits; strong numbers use factorials of digits.
Another famous strong number: 4!+0!+5!+8!+5! = 40585.
Yes. If the running factorial sum exceeds n, it cannot be strong.
State the definition, show the 0..9 lookup, then dry-run 145.
Did you Know? 🔊
Strong numbers are also called digital factorial numbers. In base 10, the classic examples are 1, 2, 145, and 40585.
Continue to Condense a Number
Learn how to repeatedly sum digits until a single digit remains.
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