Definition
3^k
Positive integers of form 3 to a power.
Check Power of 3 in Python
Overview
What You’ll Learn
Powers of three are numbers like 1, 3, 9, 27, 81… — each is 3^k for some whole k >= 0. The clearest check is: keep dividing by 3 while exact, and see if you finish at 1. This tutorial covers that loop, a live step tracer, worked Python examples, edge cases, and complexity.
Divide by 3
While exact
n //= 3 while n % 3 == 0.
End at 1
Success
Only pure powers reduce to 1.
Avoid Logs
Float risk
Integer loop is safer than log.
Live Preview
Try 27 / 10
See each division step.
vs Power of 2
Same idea
Divisor 3 instead of 2.
Introduction
A power of three is a positive integer of the form 3^k. So 27 passes (3^3) while 6 fails — it divides by 3 once and stops at 2.
Unlike power of 2, there is no popular bitwise shortcut. Interviews usually want the integer divide-by-3 loop (and a clear rejection of n <= 0).
Why it matters?
It reinforces exact integer reasoning and pairs naturally with the power-of-2 cousin interviewers often ask next.
Key Highlights
3^k Form
1, 3, 9, 27, 81…
Divide Loop
Exact // 3 until stuck.
n > 0
Zero and negatives fail.
Prefer Integers
Skip float log checks.
In short: while divisible by 3, divide; succeed only if you reach 1.
Foundation
📝 Problem & Approach
Given an integer n, decide whether it is a power of three (3^k for some k >= 0).
python
# 27 -> 27/3=9, 9/3=3, 3/3=1 -> yes
# 9 -> 9/3=3, 3/3=1 -> yes
# 6 -> 6/3=2, stop at 2 -> no
# 1 -> already 1 -> yes (3^0)Inputs & Outputs
| Item | Type | Description |
|---|---|---|
n | int | Value to test (must be > 0 for yes). |
| Return | bool | True when n is 3^k. |
| Key loop | modulo / floor | while n % 3 == 0: n //= 3 |
Minimal workflow
Pseudocode
function is_power_of_three(n):
if n <= 0:
return false
while n mod 3 == 0:
n = n / 3
return n == 1Method comparison
| Method | Idea | Notes |
|---|---|---|
| Divide by 3 | Peel factors until stuck; check == 1 | Interview default — exact |
| Generate 3^k | Multiply up and compare | Great for listing powers |
| log base 3 | Check if log is whole | Float rounding risk — avoid |
Cheat Sheet
⚡ Quick Reference
| Goal | Pattern |
|---|---|
| Reject non-positive | if n <= 0: return False |
| Divisible by 3? | n % 3 == 0 |
| Peel a factor | n //= 3 |
| Success | return n == 1 |
| Generate next | p *= 3 |
| vs power of 2 | Same loop; divisor 2 instead of 3 |
Compare
📋 Divide vs Generate vs Log
Same question — different reliability.
Divide loop
n //= 3Clearest exact check
Generate
p *= 3Best for listing powers
log3
avoid floatsRounding can lie
vs power of 2
no & trickUsually loop-only here
Context
When This Problem Shows Up
Reach for a power-of-three check whenever you need an exact 3^k test.
Interview cousins
Often asked right after power of 2.
Factor peeling
Practice exact integer division loops.
Range filtering
List 3^k values inside a band.
Teaching float traps
Contrast logs with integer loops.
Not for “divisible by 3?”
One % 3 check is a different problem.
Key benefit: one short loop that proves exactness without floating-point surprises.
Interactive
🔮 Live Preview
Shows each division step until the value is no longer divisible by 3.
Hands-On
Examples Gallery
Three complete Python programs — check 27, list powers from 1 to 20, and generate powers by multiplying. Click View Output to reveal sample console results.
📚 Getting Started
The classic divide-until-one helper.
Example 1 — Check a Single Number
Checks whether a number is exactly 3^k.
python
def is_power_of_3(n: int) -> bool:
if n <= 0:
return False
while n % 3 == 0:
n //= 3
return n == 1
number = 27
if is_power_of_3(number):
print(f"{number} is a power of 3.")
else:
print(f"{number} is not a power of 3.")How It Works
27 becomes 9, then 3, then 1. Because the final value is 1, the helper returns True.
⚡ Hunting in a Range
Reuse the helper to list nearby powers of three.
Example 2 — Powers of 3 from 1 to 20
Reuses helper function for range listing.
python
def is_power_of_3(n: int) -> bool:
if n <= 0:
return False
while n % 3 == 0:
n //= 3
return n == 1
start, end = 1, 20
print(f"Power of 3 in the range {start} to {end}:")
for i in range(start, end + 1):
if is_power_of_3(i):
print(i, end=" ")How It Works
Within 1..20 the powers are 1, 3, and 9. 27 is the next power and sits just outside this band.
Example 3 — Generate Powers by Multiplying
Build 3^k directly instead of filtering every integer.
python
print("Powers 3^k for k = 0 to 5:")
power = 1
for k in range(0, 6):
print(f"3^{k} = {power}")
power *= 3How It Works
When you only need the sequence, multiplying by 3 is cheaper than testing every n in a large range.
🧠 How the Algorithm Decides
1
Require n > 0
Zero and negatives are not powers of three here.
Guard
2
Divide while divisible
While n % 3 == 0, set n = n // 3.
Loop
3
Check the leftover
n == 1 means only factors of 3 were present.
Rule
=
Return the verdict
True for 3^k, else False.
Trace
🔎 Worked Walkthrough — 27 vs 6
Compare the divide loop on a yes case and a no case.
| Start | Steps | Stops at | Verdict |
|---|---|---|---|
27 | 27 → 9 → 3 → 1 | 1 | Yes |
9 | 9 → 3 → 1 | 1 | Yes |
6 | 6 → 2 | 2 | No |
1 | (no division) | 1 | Yes (3^0) |
Ending at 1 means every factor peeled was a 3.
Use Cases
Where power-of-three checks show up beyond the interview prompt.
1. Interview Classics
Exact integer factor peeling.
Example: is_pow3(27).
2. Range Filtering
List 3^k inside a band.
Example: 1, 3, 9 in 1..20.
3. Sequence Generation
Build powers with *= 3.
Example: Example 3.
4. Contrast With Pow2
Same loop idea, different divisor.
Example: related topic.
5. Avoiding Logs
Teach float-rounding traps.
Example: prefer // loop.
6. Next: Composite
Continue the interview chain.
Example: related CTA.
Pro Tip: say “I’ll peel factors of 3 until I can’t, then check for 1” before coding.
Advantages
Why the divide-by-3 loop works well for beginners and interviews.
1. Easy to Trace
Dry-run 27 on paper in a few steps.
2. Exact Integers
No float rounding surprises from logs.
3. Mirrors Power of 2
Same pattern with a different divisor.
4. Generates Cleanly
*= 3 builds the sequence without scanning.
Pro Tip: mention that power-of-2 has a bitwise shortcut, but power-of-3 usually stays with the loop.
Usage Tips
Small habits that keep power-of-three solutions interview-ready.
1. Guard n > 0
Reject zero and negatives first.
2. Use Floor Division
Prefer
n //= 3over float division.3. Dry-Run 27 and 6
One yes and one no seals understanding.
4. Generate When Listing
Use *= 3 if you need the sequence itself.
5. Skip Float Logs
Mention the trap, then use integers.
Pro Tip: sanity-check 1, 27, 6, and 0 — if those four behave, your logic is solid.
Common Pitfalls
Mistakes that commonly break power-of-three programs.
1. Accepting Zero
0 is divisible by 3 forever in careless loops.
→ Require n > 0 before dividing.
2. Trusting log3
Floating roots can round incorrectly.
→ Prefer the integer divide loop.
3. Stopping After One Division
Checking only n % 3 == 0.
→ Keep dividing until you cannot.
4. Forgetting 1
1 = 3^0 is a power of three.
→ Include 1 in yes cases.
5. Using Float Division
n /= 3can leave floats.→ Use
n //= 3.
Edge Cases
Handle these before claiming the check is complete.
n = 1
Yes — 3^0
Already at 1; no divisions needed.
n = 0
No for this page
n <= 0 fails immediately.
n = 6
Divisible once, then no
Stops at 2, not 1.
Negative
Not accepted
Return false here.
27
Classic yes
27 → 9 → 3 → 1.
10
Classic no
Not divisible by 3 at all.
Theory
⚖️ Facts Worth Knowing
Handy follow-ups interviewers sometimes ask.
- Pure factors. A power of 3 has no prime factors other than 3.
- Growth. Sequence: 1, 3, 9, 27, 81, 243…
- No common bit trick. Unlike 2^k, 3^k usually uses division.
- Bounded sets. For small ranges, precompute powers and test membership.
Drill
🎯 Practice Problems
Try these variations to lock in the pattern.
1. Prove 27
- Trace 27 → 9 → 3 → 1
- Confirm True
2. Reject 6
- Show stop at 2
- Confirm False
3. List 1..20
- Reproduce Example 2
- Expect 1, 3, 9
4. Generate 3^k
- Print k = 0..5
- Match Example 3
Important
Notes
- Definition: power of three means n reduces to 1 by exact division by 3.
- Check: while
n % 3 == 0, don //= 3; thenn == 1. - Integer-safe: the division loop avoids floating-point issues from logarithms.
- For bounded ranges, precompute powers of 3 and check membership. Power-of-2 has a bitwise shortcut; power-of-3 usually does not.
Quick Takeaway: peel factors of 3 with integer division; succeed only if you finish at 1.
Analysis
⏱️ Time and Space Complexity
| Approach | Time | Extra space |
|---|---|---|
| Divide-by-3 loop | O(log n) | O(1) |
| Generate up to bound | O(log bound) | O(1) or list |
| Range 1..U scan | O(U log U) worst | O(1) |
Each successful division shrinks n by a factor of 3, so the loop is logarithmic in n.
🎉 Conclusion
A power of three reduces to 1 by exact repeated division by 3. Guard n > 0, loop with n //= 3 while divisible, and prefer integers over logarithms.
Practice the three examples above, then continue to composite numbers.
Divide by 3 while exact; n == 1 means power of three.
Pro Tips
💡 Best Practices
✅ Do
- Require n > 0
- Use n //= 3 while divisible
- Succeed only when leftover is 1
- Treat 1 as yes (3^0)
- Prefer integers over log
❌ Don’t
- Accept 0 as a power of 3
- Trust float log alone
- Stop after one % 3 check
- Use /= and create floats
- Forget negatives fail
Summary
Key Takeaways
Knowledge Unlocked
Five things to remember about powers of three
Decide 3^k the interview-friendly way.
5
Core concepts 01
Rule
end at 1
Definition 02
Loop
n //= 3
Method 03
Guard
n > 0
Edges 04
Generate
p *= 3
Listing 05
Cost
O(log n)
Analysis❓ Frequently Asked Questions
A number of the form 3^k for whole k >= 0. Examples: 1, 3, 9, 27.
Yes. 3^0 = 1.
Powers of 3 are made only of factor 3. If repeated division ends at 1, it is a pure power of 3.
This tutorial returns false for n <= 0.
Yes, same repeated-division idea; only divisor changes from 2 to 3.
Possible but floating rounding can fail for large values; integer division loop is safer.
Yes. 27 / 3 = 9, 9 / 3 = 3, 3 / 3 = 1.
No. 6 is divisible by 3 once, then stops at 2, not 1.
Not a common one. Power-of-3 usually uses the divide loop.
Did you Know? 🔊
Powers of three grow as 1, 3, 9, 27, 81... Repeated division by 3 is the clearest exact integer check for this pattern.
Continue to Composite Number
Learn how to check whether a number is composite in Python.
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