What Is an Armstrong Number? — Definition, Examples and List

For an n-digit number with digits d₁, d₂, ..., dₙ:

Armstrong number condition: d₁ⁿ + d₂ⁿ + ... + dₙⁿ = the number itself

Examples: • 3-digit: each digit is cubed (power 3) • 4-digit: each digit is raised to power 4 • 1-digit: each digit is raised to power 1 (all single-digit numbers are Armstrong numbers)

Step to check:

1. Count the number of digits (n)
2. Raise each digit to the power n
3. Add all results
4. If the sum equals the original number → Armstrong number

There are exactly four 3-digit Armstrong numbers:

153: → 1³ + 5³ + 3³ = 1 + 125 + 27 = 153 ✓

370: → 3³ + 7³ + 0³ = 27 + 343 + 0 = 370 ✓

371: → 3³ + 7³ + 1³ = 27 + 343 + 1 = 371 ✓

407: → 4³ + 0³ + 7³ = 64 + 0 + 343 = 407 ✓

Verification that 100 is NOT an Armstrong number: 1³ + 0³ + 0³ = 1 + 0 + 0 = 1 ≠ 100 ✗

Verification that 200 is NOT an Armstrong number: 2³ + 0³ + 0³ = 8 ≠ 200 ✗

All single-digit numbers (0–9) are Armstrong numbers:

0¹ = 0 ✓ 1¹ = 1 ✓ 2¹ = 2 ✓ 3¹ = 3 ✓ 4¹ = 4 ✓ 5¹ = 5 ✓ 6¹ = 6 ✓ 7¹ = 7 ✓ 8¹ = 8 ✓ 9¹ = 9 ✓

This is because for a 1-digit number, the power is 1, and any digit raised to the power 1 equals itself.

There are exactly three 4-digit Armstrong numbers:

1634: → 1⁴ + 6⁴ + 3⁴ + 4⁴ = 1 + 1296 + 81 + 256 = 1634 ✓

8208: → 8⁴ + 2⁴ + 0⁴ + 8⁴ = 4096 + 16 + 0 + 4096 = 8208 ✓

9474: → 9⁴ + 4⁴ + 7⁴ + 4⁴ = 6561 + 256 + 2401 + 256 = 9474 ✓

Note: There are NO 2-digit Armstrong numbers. The 2-digit candidates would need d₁² + d₂² = the number, and no such 2-digit number exists.

Step-by-step method:

Example: Check if 371 is an Armstrong number.

Step 1: Count the digits → 371 has 3 digits, so n = 3 Step 2: Extract digits → 3, 7, 1 Step 3: Raise each to power n = 3: 3³ = 27 7³ = 343 1³ = 1 Step 4: Add: 27 + 343 + 1 = 371 Step 5: Compare with original: 371 = 371 ✓ → Armstrong number

Example: Check if 123 is an Armstrong number. Step 1: n = 3 Step 2: Digits = 1, 2, 3 Step 3: 1³ + 2³ + 3³ = 1 + 8 + 27 = 36 Step 4: 36 ≠ 123 ✗ → NOT an Armstrong number