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

Understand the Question

The smallest number that is a common multiple of a given set of numbers is their Least Common Multiple (LCM).
To find the LCM, write each number as a product of its prime factors, then multiply together the highest power of every prime factor involved.
In Grade 6, we found the smallest number divisible by all numbers from $1$ to $10$ (including $7$ ), which is $2520$ . Here, $7$ is excluded from the list.

Step 1 · Prime Factorisation

Find the prime factors for each number from $1$ to $10$ (excluding $7$ ): Diagram 1

\begin{aligned} 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 \end{aligned}

Step 2 · Calculate the LCM

To find the LCM, take the highest power of each prime factor ( $2, 3, 5$ ):

\begin{aligned} \text{LCM} &= 2^3 \times 3^2 \times 5^1 \\ &= 8 \times 9 \times 5 \\ &= 72 \times 5 \\ &= 360 \end{aligned}

Step 3 · Recall Answer from Grade 6 (Including 7)

In Grade 6, the problem included all numbers from $1$ to $10$ , which includes the prime factor $7$ ( $7^1$ ):

\begin{aligned} \text{LCM} &= 2^3 \times 3^2 \times 5^1 \times 7^1 \\ &= 8 \times 9 \times 5 \times 7 \\ &= 360 \times 7 \\ &= 2520 \end{aligned}

Answer

The smallest multiple is 360 (and from Grade 6, including 7, the answer is 2520).

Common Mistakes

Multiplying All Numbers Directly: Multiplying $1 \times 2 \times 3 \times \dots \times 10$ gives a common multiple, but not the smallest (least) common multiple.
Missing Highest Powers: Taking lower powers instead of highest powers (e.g., using $2^1$ or $2^2$ instead of $2^3$ ) when calculating the LCM.
Overlooking the Missing 7: Forgetting that $7$ is omitted from the given list, which reduces the LCM from $2520$ to $360$ .

Question 21