Find the smallest number which should be added to the smallest number divisible by 6, 9 and 15 to make it a perfect square.
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Finding the Smallest Number to Make LCM a Perfect Square
The problem asks us to find the smallest number that must be added to the smallest number divisible by 6, 9, and 15 to make it a perfect square. Let's break this down into steps.
Step 1: Find the Smallest Number Divisible by 6, 9, and 15
The smallest number divisible by 6, 9, and 15 is their Least Common Multiple (LCM). To find the LCM, we can use prime factorization.
- Prime factorization of 6: \(6 = 2 \times 3\)
- Prime factorization of 9: \(9 = 3 \times 3 = 3^2\)
- Prime factorization of 15: \(15 = 3 \times 5\)
To find the LCM, we take the highest power of each prime factor that appears in any of the numbers:
\(LCM(6, 9, 15) = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 = 90\)
So, the smallest number divisible by 6, 9, and 15 is 90.
Step 2: Find the Smallest Perfect Square Greater Than or Equal to 90
A perfect square is a number that can be obtained by squaring an integer (e.g., 1, 4, 9, 16, 25, ...). We need to find the smallest perfect square that is 90 or larger.
Let's look at the squares of integers:
- \(8^2 = 64\) (Less than 90)
- \(9^2 = 81\) (Less than 90)
- \(10^2 = 100\) (Greater than or equal to 90)
- \(11^2 = 121\) (Greater than 90, but 100 is smaller)
The smallest perfect square that is greater than or equal to 90 is 100.
Step 3: Find the Number to be Added to 90 to Make it 100
We have the number 90, and we want to reach the smallest perfect square which is 100. The number that needs to be added is the difference between the perfect square and 90.
Number to be added = Smallest perfect square \(\ge 90\) - 90
Number to be added = \(100 - 90 = 10\)
Therefore, the smallest number that should be added to 90 to make it a perfect square (100) is 10.
Revision Table: Key Concepts
| Concept | Description | Example |
|---|---|---|
| Least Common Multiple (LCM) | The smallest positive integer that is a multiple of two or more numbers. | LCM(4, 6) = 12 |
| Prime Factorization | Expressing a number as a product of its prime factors. | 12 = \(2^2 \times 3\) |
| Perfect Square | An integer that is the square of an integer. | 25 (since \(5^2 = 25\)) |
Additional Information: Understanding Perfect Squares and LCM
Understanding LCM and perfect squares is crucial for solving problems like this. The LCM represents a common point or cycle for multiple numbers, while perfect squares are numbers with specific properties related to area (as a square with integer sides) or factorization (all prime factors have even exponents).
When you need to find the smallest number to add to a given number 'N' to make it a perfect square, you first find the smallest perfect square 'S' that is greater than or equal to 'N'. Then, the number to add is 'S - N'.
In our problem, the given number 'N' was the LCM of 6, 9, and 15, which is 90. The smallest perfect square 'S' greater than or equal to 90 is 100. So, we add \(100 - 90 = 10\).
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select the correct answer using the code given below:
