Introduction to Perfect Squares and Factorization

In mathematics, a perfect square is a number that can be expressed as the product of an integer with itself. In this article, we will determine the smallest integer value of k such that 150k is a perfect square. This problem involves understanding and applying fundamental concepts of number theory and algebra, including integer factorization and perfect squares. Let's explore this in detail.

Factorizing 150

Firstly, we need to factorize the number 150 into its prime factors:

150 2 times; 3 times; 52

For 150k to be a perfect square, all the exponents in its prime factorization must be even. The current exponents are:

21 31 52

Since the exponents of 2 and 3 are currently 1 (which is odd), we need to adjust them to make them even. Specifically:

For 21, we need another 2 to make 22. For 31, we need another 3 to make 32.

Therefore, k must include these extra factors:

k 2 times; 3 6

Verification and Generalization

To verify, we can substitute k 6 into the equation:

150k 150 times; 6 900 302

Since 900 is a perfect square, our solution is correct. The smallest integer k that satisfies the condition is 6.

Similar Problems and Solutions

Consider another similar problem: if we have the equation 150k m2, we can derive:

2 times; 3 times; 52 times; k 22 times; 32 times; 55

From this, we can see that the smallest possible integer value of k is:

k 6

Another way to approach the problem is by expressing 150 as:

150 27 times; 5

To make 150k a perfect square, k must complete the square by adding the missing factors:

k 2 times; 3 6

Conclusion and Future Directions

In conclusion, the smallest integer value of k such that 150k is a perfect square is 6. Understanding and applying the concept of factorization and perfect squares is crucial in solving such problems. This knowledge can be extended to more complex scenarios involving larger numbers and more complex equations.