Find the smallest integer value for which 240k is a perfect square
To find the smallest integer value for which 240k is a perfect square, we need to find a value of k that makes 240k a perfect square.
Let's start by factoring out the common factors of 240: 2 * 120 = 2 * 2 * 60 = 2^2 * 2 * 2 * 30.
Now we can see that the prime factorization of 240 contains two 2s, which means it must have an even exponent in the perfect square. Let's look at the exponents of the other prime factors: 2^2 * (2 * 30).
To make the exponent of 2 even, we need to multiply it by 2. However, this is not enough to make the entire expression a perfect square because the exponent of 30 is 1, which is odd.
To make the exponent of 30 even, we need to multiply it by 30. This gives us 2^2 * 30^2.
Therefore, the smallest integer value for which 240k is a perfect square is k = 30^2 = 900.
So, 240 * k = 240 * 900 = 216,000 is a perfect square.
240k = 2*2*2*2*15 k = 2^2 * 2^2 * 3 * 5
so k better be 3*5 = 15
try that
240 * 15 = 3600 = 6^2 * 10^2 = 60^2
that works
You are correct. Thank you for pointing out the error in my previous response. The smallest integer value for which 240k is a perfect square is k = 15.
To find the smallest integer value for which 240k is a perfect square, we need to determine the factors of 240 and see if any of them can be multiplied by k to make a perfect square.
Let's find the prime factorization of 240:
240 = 2^4 * 3 * 5
Now, we need to consider the exponents of the prime factors. Since we want the result to be a perfect square, each exponent should be an even number.
Let's check the possible combinations for the exponents:
1. 2^0 * 3^0 * 5^0 = 1 (not possible)
2. 2^2 * 3^0 * 5^0 = 4 (possible)
3. 2^0 * 3^2 * 5^0 = 9 (possible)
4. 2^2 * 3^2 * 5^0 = 36 (possible)
5. 2^0 * 3^0 * 5^2 = 25 (not possible)
6. 2^2 * 3^0 * 5^2 = 100 (possible)
7. 2^0 * 3^2 * 5^2 = 225 (possible)
8. 2^2 * 3^2 * 5^2 = 900 (possible)
From the above combinations, we can see that the smallest possible perfect square is when k = 100, which gives us 240k = 240 * 100 = 24000.
Therefore, the smallest integer value for which 240k is a perfect square is 24000.