\textLCM(15, 25) = 3 \cdot 5^2 = 3 \cdot 25 = 75 - Dachbleche24
Understanding LCM(15, 25) = 75: The Full Prime Factorization Explained
Understanding LCM(15, 25) = 75: The Full Prime Factorization Explained
When solving math problems involving least common multiples (LCM), understanding the underlying number theory is key. One commonly encountered example is finding LCM(15, 25), which equals 75—but what does that truly mean, and how is it derived?
In this guide, we break down LCM(15, 25) using prime factorization to reveal the full reasoning behind why the least common multiple is 3 × 5² = 75. Whether you're a student, educator, or math enthusiast, this explanation will deepen your grasp of LCM and its connection to prime factors.
Understanding the Context
What is LCM?
The least common multiple (LCM) of two or more integers is the smallest positive integer that is divisible by each of them. For example, the multiples of 15 are: 15, 30, 45, 75, 90, ... and multiples of 25 are: 25, 50, 75, 100, .... The smallest shared multiple is 75—confirming LCM(15, 25) = 75.
But why does this number—3 × 5²—carry such significance?
Key Insights
Step-by-Step: Prime Factorization of 15 and 25
To compute LCM, we begin by factoring both numbers into their prime components:
- 15 = 3 × 5
- 25 = 5 × 5 = 5²
These prime factorizations reveal the “building blocks” of each number. The LCM is formed by taking each prime factor raised to its highest exponent appearing across the factorizations.
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How to Compute LCM Using Prime Exponents
Given:
- 15 = 3¹ × 5¹
- 25 = 5²
Now, identify each prime and take the highest exponent:
| Prime | Max Exponent in 15 | Max Exponent in 25 | In LCM |
|-------|---------------------|--------------------|--------|
| 3 | 1 | 0 | 3¹ |
| 5 | 1 | 2 | 5² |
Multiply these together:
LCM(15, 25) = 3¹ × 5²
Simplify the Expression
We simplify:
5² = 25, so:
3 × 25 = 75
Thus, LCM(15, 25) = 75 — expressed compactly as 3 × 5² = 75.
