What is the least perfect square which is divisible by 3, 4, 5, 6 and 7?
44100
Finding the Least Perfect Square Divisible by Multiple Numbers
The problem asks for the least perfect square number that is divisible by 3, 4, 5, 6, and 7. For a number to be divisible by multiple numbers, it must be a multiple of their Least Common Multiple (LCM).
First, let's find the LCM of 3, 4, 5, 6, and 7. We can do this by finding the prime factorization of each number:
- Prime factorization of 3: \(3^1\)
- Prime factorization of 4: \(2^2\)
- Prime factorization of 5: \(5^1\)
- Prime factorization of 6: \(2^1 \times 3^1\)
- Prime factorization of 7: \(7^1\)
The LCM is found by taking the highest power of all prime factors that appear in any of the factorizations:
LCM(3, 4, 5, 6, 7) = \(2^2 \times 3^1 \times 5^1 \times 7^1\)
Calculating the value:
LCM = \(4 \times 3 \times 5 \times 7 = 12 \times 35 = 420\)
So, the smallest number divisible by 3, 4, 5, 6, and 7 is 420.
Now, we need to find the least perfect square that is a multiple of 420. A perfect square is a number whose prime factorization has only even exponents. Let's look at the prime factorization of 420 again:
\(420 = 2^2 \times 3^1 \times 5^1 \times 7^1\)
To make 420 a perfect square, we need to multiply it by factors that will make all the exponents even. The exponent of 2 is already even (2). The exponents of 3, 5, and 7 are 1 (odd). To make these exponents even, we need to multiply by \(3^1\), \(5^1\), and \(7^1\).
The multiplier needed is \(3^1 \times 5^1 \times 7^1 = 3 \times 5 \times 7 = 105\).
The least perfect square divisible by 3, 4, 5, 6, and 7 is the LCM multiplied by this required factor:
Least Perfect Square = \(420 \times 105\)
Calculation:
\(420 \times 105 = 420 \times (100 + 5) = 420 \times 100 + 420 \times 5 = 42000 + 2100 = 44100\)
Let's verify if 44100 is a perfect square:
\(44100 = 441 \times 100 = 21^2 \times 10^2 = (21 \times 10)^2 = 210^2\)
Yes, 44100 is the square of 210, so it is a perfect square.
Since 44100 is a multiple of 420 (which is the LCM of 3, 4, 5, 6, and 7), it is divisible by all these numbers. It is the smallest such perfect square because we multiplied the LCM by the minimum necessary factors to make the exponents even.
Revision Table: Key Concepts
| Concept | Explanation | How it applies here |
|---|---|---|
| Least Common Multiple (LCM) | The smallest positive integer that is a multiple of two or more integers. | Finding the smallest number divisible by 3, 4, 5, 6, and 7 requires calculating their LCM. |
| Prime Factorization | Expressing a number as a product of its prime factors. | Used to find the LCM and to determine if a number is a perfect square. |
| Perfect Square | An integer that is the square of an integer (e.g., 9 is a perfect square because \(9 = 3^2\)). The exponents in its prime factorization are always even. | The final answer must be a number that is a perfect square. |
Additional Information: Perfect Squares and Divisibility
A number is a perfect square if, in its prime factorization, the power of every prime factor is an even number. For example, \(36 = 2^2 \times 3^2\). Both exponents (2 and 2) are even, so 36 is a perfect square (\(6^2\)).
When you have the prime factorization of the LCM and want to find the smallest multiple that is a perfect square, you look at the exponents of each prime factor in the LCM's factorization. If an exponent is odd, you need to multiply the LCM by that prime factor raised to the power of 1 to make the exponent even (e.g., \(p^3\) becomes \(p^{3+1} = p^4\) by multiplying by \(p^1\)). If an exponent is already even, you don't need to multiply by that prime factor.
In our case, LCM = \(2^2 \times 3^1 \times 5^1 \times 7^1\). The exponents for 3, 5, and 7 are 1 (odd). We need to multiply by \(3^1\), \(5^1\), and \(7^1\) to make their exponents 2. The exponent for 2 is already 2 (even), so we don't need to multiply by any more factors of 2.
The smallest perfect square multiple is therefore \(2^2 \times 3^{1+1} \times 5^{1+1} \times 7^{1+1} = 2^2 \times 3^2 \times 5^2 \times 7^2\). This can be written as \((2 \times 3 \times 5 \times 7)^2\), which is the square of the LCM multiplied by the necessary factors.
The calculation was LCM \(\times\) \((3^1 \times 5^1 \times 7^1) = (2^2 \times 3^1 \times 5^1 \times 7^1) \times (3^1 \times 5^1 \times 7^1) = 2^2 \times 3^2 \times 5^2 \times 7^2\).
This confirms that \(44100 = (2 \times 3 \times 5 \times 7)^2 = 210^2\).
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