BIZOMAT MAKEOS · Mathematics
Prime Factors, Greatest Common Divisor and Least Common Multiple
What are the prime factors of these numbers, and their greatest common divisor and least common multiple?
Prime factorisation, greatest common divisor and least common multiple of two or three whole numbers, for gear tooth counts, hunting-tooth ratios, change gears and repeat cycles.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| First number | First whole number (for example gear teeth). | |
| Second number | Second whole number. | |
| Third number (0 = not used) | Optional third whole number. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: First number
- What it is
- The first whole number to factorise, such as the tooth count of one gear or the length of one repeat cycle.
- Units and format
- Enter a plain number with no unit. The form accepts 1 or more. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 72 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Second number
- What it is
- The second whole number, such as the tooth count of the mating gear. The hunting-tooth check uses this and the first number.
- Units and format
- Enter a plain number with no unit. The form accepts 1 or more. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Third number (0 = not used)
- What it is
- An optional third whole number included in the common divisor and common multiple, such as a third gear or cycle. Zero leaves it out.
- Units and format
- Enter a plain number with no unit. The form accepts 0 or more. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Method and formulas
- Prime factors by trial division.
- GCD by Euclid’s algorithm; LCM = a × b ÷ GCD (extended to three numbers).
- Two gears form a hunting-tooth pair when their GCD is 1 (every tooth meets every other tooth).
- Example: 72 = 2 × 2 × 2 × 3 × 3 and 30 = 2 × 3 × 5; GCD 6, LCM 360.
Limitations
- Whole numbers up to one billion.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check that a gear pair has a hunting-tooth ratio
- Factor a ratio into change gears
- Find when two cycles repeat together
Who it is for
- Gear designers
- Machinists
- Engineers
What you get
- Prime factors of each number
- Greatest common divisor
- Least common multiple
- Hunting-tooth check for two gears
Interpreting the result
- A GCD of 1 between gear tooth counts spreads wear evenly; a large common factor repeats the same tooth contacts.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 20–29. Topic reference; no source tables or recommended settings are reproduced.
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