lcm of 15 and 60

Understand the Problem

The question is asking for the least common multiple (LCM) of two numbers, which is the smallest number that is a multiple of both 15 and 60. We will find this by identifying the prime factors of both numbers or by listing their multiples.

Answer

The least common multiple (LCM) of 15 and 60 is \(60\).

Steps to Solve

Identify the Prime Factors of Each Number

First, find the prime factorization of both 15 and 60.

For 15:

The prime factors of 15 are (3) and (5).
Therefore, (15 = 3^1 \times 5^1).

For 60:

The prime factors of 60 are (2), (3), and (5).
Therefore, (60 = 2^2 \times 3^1 \times 5^1).

Determine the Highest Power of Each Prime Factor

Next, we need to take the highest power of each prime factor from both factorizations.

For the prime factor (2), the highest power is (2^2) (from 60).
For the prime factor (3), the highest power is (3^1) (common to both).
For the prime factor (5), the highest power is (5^1) (common to both).

Calculate the LCM

Now, multiply the highest powers of each prime factor together:

[ \text{LCM} = 2^2 \times 3^1 \times 5^1 ]

Calculating this:

[ \text{LCM} = 4 \times 3 \times 5 ]

First, calculate (4 \times 3 = 12) and then (12 \times 5 = 60).

State the Result

The least common multiple of 15 and 60 is the product we calculated.

The least common multiple (LCM) of 15 and 60 is (60).

More Information

The least common multiple is the smallest multiple that two numbers share. In this case, both 15 and 60 can divide 60 without leaving a remainder. LCM is useful in adding fractions and solving problems involving cycles or patterns.

Tips

Confusing LCM with GCD: Remember, LCM is about finding the smallest multiple while GCD (Greatest Common Divisor) is about the largest factor.
Forgetting to take the highest powers of prime factors when combining them.

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