Find the smallest positive integer P such that the cube root of 400 times P is an integer.
This sounds awkward, but is actually easy. Consider the number 400P. It must end in zero - actually, it must end in 00, since it has 100 as a factor.
Therefore its cube root must end in zero. Check this for yourself: the last digit of a cube can be determined by the last digit of the cube root. 1x1x1 = 2, 2x2x2=8 and so on, regardless of the higher digits.
So for the cube root we're already down to factors of 10 - we only have to look at 10, 20, 30... and their cubes all end in 000 because of the three 10s in the factors.
Does 10 work? No, 1000 is a cube of 10, but 400 doesn't divide evenly.
Now try 20.
20 * 20 = 400, so 20 is only the square root of 400, not the cube root.
Unfortunately, I can only tell you that 20 is not the answer, but I do not know the answer. Sorry.
400 = 2x2x2 x 2 x 5x5
every triplet of "same factors" will produce a cube
so to make the right side of the above statement into a perfect cube, we need another 2x2x5 or 20
So let's multiply both sides by 20
400x20 = 2x2x2 x 2x2x2 x5x5x5
400(20) = 8000
400P = 8000
so P = 20
I think the easiest is to say the
cuberoot(400P)
=cuberoot(2x2x2x2x5x5xP)
=2cuberoot(2x5x5xP)
for cuberoot(2x5x5xP) to be an integer
then P=2x2x5=20
so why is 20 not the answer?
To find the smallest positive integer P such that the cube root of 400 times P is an integer, we can follow these steps:
1. Start by finding the prime factorization of 400. The prime factorization of 400 is 2^4 * 5^2.
2. Since the cube root of 400 times P needs to be an integer, we need to make sure that the exponents of the prime factors in the prime factorization of P are divisible by 3.
3. Considering the prime factorization of 400, we see that the exponent of 2 is 4, which is not divisible by 3. Thus, we need to multiply 400 by some power of 2 to make the exponent divisible by 3.
4. The smallest power of 2 that makes the exponent of 2 divisible by 3 is 2^2. So, we multiply 400 by 2^2, which gives us 400 * 4.
5. Now, we have to consider the exponent of 5 in the prime factorization of 400. The exponent of 5 is 2, which is not divisible by 3. Therefore, we need to multiply 400 * 4 by some power of 5 to make the exponent divisible by 3.
6. The smallest power of 5 that makes the exponent of 5 divisible by 3 is 5^1. So, we multiply 400 * 4 by 5^1, which gives us 400 * 4 * 5.
7. The product of 400 * 4 * 5 is 8000.
Therefore, the smallest positive integer P such that the cube root of 400 times P is an integer is 8000.