Finding the Least Common Multiple (LCM)

The question asks us to find the Least Common Multiple (LCM) of the numbers 16, 28, and 42. The LCM is the smallest positive integer that is a multiple of all the given numbers. One common method to find the LCM is by using prime factorization.

Steps to Find the LCM using Prime Factorization

Here are the steps involved:

1. Find the prime factorization of each number.
2. For each distinct prime factor found, identify the highest power to which it is raised in any of the factorizations.
3. Multiply these highest powers together. The result is the LCM.

Prime Factorization of 16, 28, and 42

Let's find the prime factors for each number:

• Prime factors of 16:

We can divide 16 by the smallest prime number, 2, repeatedly:

$16 \div 2 = 8$
$8 \div 2 = 4$
$4 \div 2 = 2$
$2 \div 2 = 1$

So, the prime factorization of 16 is $2 \times 2 \times 2 \times 2 = 2^4$.

• Prime factors of 28:

Divide 28 by the smallest prime number, 2:

$28 \div 2 = 14$
$14 \div 2 = 7$
Now, 7 is a prime number, so we stop.

So, the prime factorization of 28 is $2 \times 2 \times 7 = 2^2 \times 7^1$.

• Prime factors of 42:

Divide 42 by the smallest prime number, 2:

$42 \div 2 = 21$
Now, 21 is not divisible by 2, but it is divisible by the next prime number, 3:

$21 \div 3 = 7$
Again, 7 is a prime number, so we stop.

So, the prime factorization of 42 is $2 \times 3 \times 7 = 2^1 \times 3^1 \times 7^1$.

Identifying Highest Powers

Now, let's look at the prime factors and their highest powers across the numbers 16, 28, and 42:

Calculating the LCM

To find the LCM, we multiply the highest powers of all distinct prime factors (2, 3, and 7) we identified:

LCM = (Highest power of 2) $\times$ (Highest power of 3) $\times$ (Highest power of 7)

LCM = $2^4 \times 3^1 \times 7^1$

LCM = $16 \times 3 \times 7$

LCM = $48 \times 7$

LCM = $336$

The Least Common Multiple of 16, 28, and 42 is 336.

Revision Table: LCM and HCF Basics

Additional Information on LCM Calculation

Besides prime factorization, another method to find the LCM is listing multiples:

• List multiples of 16: 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240, 256, 272, 288, 304, 320, 336, ...
• List multiples of 28: 28, 56, 84, 112, 140, 168, 196, 224, 252, 280, 308, 336, ...
• List multiples of 42: 42, 84, 126, 168, 210, 252, 294, 336, ...

The smallest number that appears in all three lists is 336. This confirms our result from the prime factorization method.

The relationship between LCM and HCF for two numbers 'a' and 'b' is given by: $a \times b = \text{LCM}(a, b) \times \text{HCF}(a, b)$. This relationship does not directly extend to three numbers in the same simple product form, but prime factorization remains a reliable method for any number of integers.