What is the smallest number that is a multiple of 1, 2, 3, 4, 5, 6, 8, 9, 10? Do you remember the answer from Grade 6, Chapter 5?

Question

Homi · Accepted Answer

We need to find the Least Common Multiple (LCM) of the given numbers.

Step 1 — Prime Factorization

Let us find the prime factors for each number. Prime factors are prime numbers. They multiply to make the original number. We list them for numbers 1 to 10. We skip number 7 for now.

$1 = 1$

$2 = 2$

$3 = 3$

$4 = 2 \times 2$

$5 = 5$

$6 = 2 \times 3$

$8 = 2 \times 2 \times 2$

$9 = 3 \times 3$

$10 = 2 \times 5$

Diagram 1

Step 2 — Calculating the LCM

To find the LCM, we use prime factors. We take the highest power of each one. The prime factors we found are 2, 3, and 5. Let us find the highest power for each prime factor. The highest power of 2 is $2 \times 2 \times 2$ . This is written as $2^3$ . The highest power of 3 is $3 \times 3$ . This is written as $3^2$ . The highest power of 5 is $5^1$ . Now, we multiply these highest powers together.

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

$= 8 \times 9 \times 5$

$= 72 \times 5$

$\boxed{360}$

Step 3 — Including the number 7

The problem asks what happens if 7 is also included. If 7 is included, 7 becomes a prime factor. The highest power of 7 is $7^1$ . We multiply our previous LCM by this new prime factor.

$\text{LCM} = 2^3 \times 3^2 \times 5^1 \times 7^1$

$= 8 \times 9 \times 5 \times 7$

$= 360 \times 7$

$\boxed{2520}$

Answer

(i) The smallest number that is a multiple of 1, 2, 3, 4, 5, 6, 8, 9, 10 is 360. (ii) If 7 is also included, the smallest number is 2520.

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