To analyze the scenario where a number is the product of 6 and 225, we first calculate the product:
\[ 6 \times 225 = 1350 \]
Next, we want to determine how this number relates to perfect square multipliers.
Step 1: Factorization of 1350
We can factor 1350 to understand its prime components:
\[ 1350 = 6 \times 225 = 2 \times 3 \times 15^2 \]
Further breaking it down:
\[ 15 = 3 \times 5 \quad \Rightarrow \quad 15^2 = (3 \times 5)^2 = 3^2 \times 5^2 \]
Thus,
\[ 225 = 15^2 = 3^2 \times 5^2 \]
Combining everything:
\[ 1350 = 2 \times 3^3 \times 5^2 \]
Step 2: Expressing 1350 in terms of primes
Now, we express 1350 as:
\[ 1350 = 2^1 \times 3^3 \times 5^2 \]
Step 3: Perfect Square Multiplier
A perfect square is a number that can be expressed as \(n^2\) for some integer \(n\). The prime factorization of a perfect square must have even powers for all prime factors.
To find a perfect square multiplier for 1350, we analyze the exponents in the factorization of 1350:
- \(2^1\) (needs to be raised to \(2^2\), so we need one more \(2\))
- \(3^3\) (needs to be raised to \(3^4\), so we need one more \(3\))
- \(5^2\) (already a perfect square)
Step 4: Finding the multiplier
The necessary add-on to make the product a perfect square is:
\[ 2^1 \times 3^1 \]
Thus, a multiplier of:
\[ 2^1 \times 3^1 = 2 \times 3 = 6 \]
Step 5: Conclusion
To find a perfect square number that can be multiplied with 1350 to result in a perfect square:
\[ \text{Perfect Square Multiplier} = 6 \]
Therefore, if we multiply 1350 by its multiplier of 6:
\[ 1350 \times 6 = 8100 \quad \text{(which is a perfect square since } 90^2 = 8100) \]
In summary, 1350 is the product of 6 and 225, and for it to become a perfect square through multiplication, we multiply it by the perfect square multiplier of \(6\).
1 answer