An Armstrong number (also called a narcissistic number or pluperfect digital invariant) is a number that is equal to the sum of its own digits, each raised to the power equal to the total number of digits in the number. For a 3-digit number, each digit is raised to the power 3. For a 4-digit number, each digit is raised to the power 4 — and so on. The most well-known example is 153, because 1³ + 5³ + 3³ = 1 + 125 + 27 = 153.
Armstrong number: a number equal to the sum of its digits, each raised to the power of the number of digits.
3-digit Armstrong numbers (exactly 4): 153, 370, 371, 407.
4-digit Armstrong numbers (exactly 3): 1634, 8208, 9474.
All single-digit numbers (0–9) are Armstrong numbers.
There are NO 2-digit Armstrong numbers.
153 = 1³ + 5³ + 3³ = 1 + 125 + 27 = 153 — the most famous example.
Also called: narcissistic numbers, pluperfect digital invariants.
In programming (CBSE Class 11/12), checking for Armstrong numbers is a standard loop exercise.
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:
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
An Armstrong number is a number that equals the sum of its own digits, each raised to the power equal to the number of digits. For example, 153 is a 3-digit number: 1³ + 5³ + 3³ = 1 + 125 + 27 = 153. Armstrong numbers are also called narcissistic numbers.
There are exactly four 3-digit Armstrong numbers: 153 (1³+5³+3³=153), 370 (3³+7³+0³=370), 371 (3³+7³+1³=371), and 407 (4³+0³+7³=407).
Yes. 153 is a 3-digit number, so each digit is raised to the power 3: 1³ + 5³ + 3³ = 1 + 125 + 27 = 153. Since the sum equals the original number, 153 is an Armstrong number.
Yes. 370 has 3 digits: 3³ + 7³ + 0³ = 27 + 343 + 0 = 370. The sum equals the number, so 370 is an Armstrong number.
The four-digit Armstrong numbers are: 1634 (1⁴+6⁴+3⁴+4⁴ = 1+1296+81+256 = 1634), 8208 (8⁴+2⁴+0⁴+8⁴ = 4096+16+0+4096 = 8208), and 9474 (9⁴+4⁴+7⁴+4⁴ = 6561+256+2401+256 = 9474).
Co-prime Numbers — Definition, Examples, and Properties
Co-prime numbers are pairs of numbers whose HCF is 1. Examples: (8, 15), (4, 9), (1, n). They need not be prime individually. Learn properties and FAQs.
Value of cos 225° and sin 165°
Learn how to calculate the exact values of cos 225 and sin 165 using trigonometric identities and quadrant rules.
Find the Value of: cos(24°) + cos(55°) + cos(125°) + cos(204°)
Learn the step-by-step solution to the famous trigonometry problem: cos(24) + cos(55) + cos(125) + cos(204). Understand ASTC quadrant rules to get the answer 0.
Cos2A Formula (Double Angle Identity)
Master the cos2A formula. Learn all three variations of the double angle identity for cosine in terms of sin, cos, and tan.
What is the Value of cos 37°?
cos 37° = 4/5 = 0.8 (standard approximation). Exact value: cos 37° ≈ 0.7986. Derived from the 3-4-5 right triangle. Used in physics problems.
Turn this guide into revision flashcards, a practice exam, or an AI-generated podcast — free, no signup required.