Understanding the Smallest Square Number Divisible by 4, 8, and 18

When faced with questions about finding the smallest square number that is divisible by specific numbers, a key strategy is to use the least common multiple (LCM). An LCM ensures that the number is divisible by all given numbers, making the problem simpler to solve. In this article, we'll explore the process step-by-step, using real examples and practical methods.

Step-by-Step Solution to Find the Smallest Square Number

Step 1: Find the LCM of the numbers.

Given the numbers 4, 8, and 18, let's first calculate their LCM. The LCM is the smallest number that is exactly divisible by each of the given numbers.

LCM(4, 8, 18)  72

So, the LCM of 4, 8, and 18 is 72.

Step 2: Prime Factorize the LCM.

Next, we need to prime factorize 72 to see its prime factors:

72  23 × 32

The prime factorization of 72 reveals that 2 has an odd exponent (3) and 3 has an even exponent (2).

Step 3: Adjust the Exponents to Make the Number a Square.

For a number to be a square, all its prime factors must have even exponents. Since 3 already has an even exponent (2), we need to adjust the exponent of 2 to make it even. The nearest even number to 3 is 4 (2×2). Therefore, we multiply 72 by 2 (which is 21) to get:

72 × 2  144

Now, 144 can be written as:

144  2? × 32

144 has all even exponents, making it a perfect square. Thus, the smallest square number divisible by 4, 8, and 18 is 144.

Verification and J Programming Language Approach

To verify the result, we can use a simple example in the J programming language:

{.b~./04 8 18/b.a~. :a.1i.1000144

The J programming language solution confirms that 144 is the smallest square number divisible by 4, 8, and 18.

We can also manually check whether 144 is divisible by 4, 8, and 18:

% 144124 8 18/1440 0 0

144 is indeed divisible by 4, 8, and 18 with no remainders, verifying our solution.

Similar Problem: Smallest Square Divisible by 4, 8, 18, 27, and 50

Let's extend our understanding to a similar but more complex problem: finding the smallest square number divisible by 4, 8, 18, 27, and 50. We'll follow a systematic approach:

Step 1: Prime Factorization

List the prime factors of each number:

4 2 × 2 22 8 2 × 2 × 2 23 18 2 × 3 × 3 21 × 32 27 3 × 3 × 3 33 50 2 × 5 × 5 21 × 52

Step 2: Identify the Largest Exponents

For each prime factor, we need to identify the largest exponent:

For 2: Largest is 23 (from 8) For 3: Largest is 33 (from 27) For 5: Largest is 52 (from 50)

Step 3: Adjust Odd Exponents to Make Them Even

Add one to any odd exponent to make the number a square:

23 changes to 2? (2 × 2 × 2 × 2) 33 changes to 3? (3 × 3 × 3 × 3) 52 remains 52 (since it is already even)

Step 4: Multiply the Prime Factors with Their New Exponents

2? × 3? × 52  16 × 81 × 25  32400

The smallest square number that is divisible by 4, 8, 18, 27, and 50 is 32400.

Your Turn: Practice with a Similar Problem

Try solving a similar problem on your own. Use the steps we've outlined to find the smallest square number that is divisible by a set of given numbers. Here's an example:

Find the smallest square number divisible by 6, 12, and 18.

Remember, prime factorization, identifying the largest exponents, adjusting any odd exponents, and then multiplying the prime factors with their new exponents will lead you to the answer.