Number dictionary · Exact arithmetic
138: factors and number facts
one hundred thirty-eight is an even number and is composite. Its 8 positive divisors form 4 factor pairs.
Number JD Editorial Team · Updated 2026-10-08
- In words
- one hundred thirty-eight
- Roman numeral
- CXXXVIII
- Prime factorization
- 2 × 3 × 23
- Digital root
- 3
All factors of 138
A positive factor divides 138 with no remainder. In ascending order, the complete list is:
1, 2, 3, 6, 23, 46, 69, 138
Factor pairs
- 1 × 138 = 138
- 2 × 69 = 138
- 3 × 46 = 138
- 6 × 23 = 138
Each pair describes an arrangement of 138 items into equal rows. The pairs have different row and column counts; 138 cannot make a square array of whole items.
Divisor totals
- Positive divisors
- 8
- Sum of all divisors
- 288
- Sum excluding 138
- 150
- Classification
- abundant
138 is abundant because its proper divisors add to 150, which is greater than 138.
Prime factorization of 138
138 = 2 × 3 × 23
Repeated division, step by step
- 138 ÷ 2 = 69
- 69 ÷ 3 = 23
- 23 ÷ 23 = 1
Multiply the prime divisors in these steps to reconstruct 138. Each prime in this factorization occurs once. The divisor-count calculation is (1 + 1) × (1 + 1) × (1 + 1) = 8.
The preceding prime is 137. The next larger prime is 139. The factor list above explains why 138 is composite.
Learn the factorization method138 in words, Roman numerals and other bases
| Representation | Value |
|---|---|
| English (US style) | one hundred thirty-eight |
| Roman numeral | CXXXVIII |
| Binary (base 2) | 10001010 |
| Octal (base 8) | 212 |
| Decimal (base 10) | 138 |
| Hexadecimal (base 16) | 8A |
Check the binary conversion
Read 10001010 from right to left using place values 1, 2, 4, 8 and successive powers of two. Keep the values whose bit is 1:
128 + 8 + 2 = 138
Check the hexadecimal conversion
8 × 16^1 + 10 × 16^0 = 138
Hexadecimal uses A through F for digit values 10 through 15. These notations represent the same quantity; changing the base does not change the integer.
Square, cube and square root of 138
- 138 squared
- 138 × 138 = 19,044
- 138 cubed
- 138 × 138 × 138 = 2,628,072
- Square root
- ≈ 11.747340
- Simplified radical
- √138
138 lies between 121 (11²) and 144 (12²), so it is not a perfect square. It is not a perfect cube.
138 is not triangular: no positive whole-number k satisfies k(k + 1) / 2 = 138.
Distinguish a square from a square rootMultiplication table for 138
These are the first twelve positive multiples. Unlike the finite factor list, the sequence of multiples continues without an upper limit.
Divisibility checks
| Divide by | Whole quotient | Remainder | Divides exactly? |
|---|---|---|---|
| 2 | 69 | 0 | Yes |
| 3 | 46 | 0 | Yes |
| 4 | 34 | 2 | No |
| 5 | 27 | 3 | No |
| 6 | 23 | 0 | Yes |
| 7 | 19 | 5 | No |
| 8 | 17 | 2 | No |
| 9 | 15 | 3 | No |
| 10 | 13 | 8 | No |
| 11 | 12 | 6 | No |
| 12 | 11 | 6 | No |
For example, 138 = 7 × 19 + 5. A remainder of zero is what makes a divisor a factor.
Digit sum and digital root of 138
138 contains 3 decimal digits. Its digit sum is 1 + 3 + 8 = 12.
- 138: 1 + 3 + 8 = 12
- 12: 1 + 2 = 3
The digital root is 3. Reversing its digits gives 831; it is not a decimal palindrome. Digit patterns depend on the notation being used.
Digital-root arithmetic always finishes at a single digit for a positive integer. Some numerology methods stop at 11, 22 or 33 instead; that is a separate interpretive convention.
Fractions and a worked comparison
As a fraction of 100, 138/100 simplifies to 69/50 by dividing both terms by their greatest common divisor, 2. Its decimal value is 1.38. Because the numerator exceeds the denominator, this value is greater than one.
The greatest common divisor of 138 and 12 is 6. Their least common multiple is 276. The identity GCD × LCM = 138 × 12 = 1656 checks both results.
Work through GCF and LCMQuestions about 138
What are the factors of 138?
The positive factors are 1, 2, 3, 6, 23, 46, 69, 138. There are 8 factors, including 1 and 138.
Is 138 a prime number?
No. 138 is composite: 2 × 3 × 23 = 138.
How do you write 138 in Roman numerals?
138 is CXXXVIII in standard subtractive Roman numeral notation.
What is the binary representation of 138?
138 in decimal is 10001010 in base 2. Adding its active place values gives 128 + 8 + 2 = 138.
Methods and reference definitions
Number JD calculates each result from the integer on this page. Positive factors are enumerated in pairs; repeated division gives the prime factorization; place values verify base conversions. Decimal square roots are rounded to six places when irrational.
NUMBER JOURNAL / JD
A reflective note on 138
Meaning varies by tradition and personal experience. One sighting is not evidence of what will happen next.
The digits of 138 add to 12. That arithmetic result is separate from the cultural symbolism of the pattern. If this number matters to you, write down what you noticed and choose one grounded action rather than treating the sighting as a prediction.