Let $N=2^{10}\cdot 3^5$. What is the smallest positive integer whose product with $N$ is a perfect square?

The smallest positive integer is 3.

Step-by-step explanation:

In order to detect if a number is a perfect square, you firstly have to write the number as a product of its prime factor. If the prime factors of the number appear in an even number of times, then it's a perfect square. For example, if we want to detect if 36 is a perfect square we have to write 36 as a product of its prime factors. Doing this, we have:

36 = 2 x 2 x 3 x 3 = 2² x 3². From here, we see that the prime factors of 36 – 2 and 3 – appear twice which is an even number of times. Hence, it's a perfect square.

Now let us go back to our number:

N = 2^10 x 3^5

Obviously, N is not a perfect square because one of it's prime factors, 3, appears 5 times which is an odd number of times. In order to make it a perfect square, we have to multiply it by a certain number of 3's such that it will appear in an even number of times. Since the least positive integer is required, then we have to multiply by the least number of 3 that will make 3^5 even. Obviously, one 3 is the least needed to make 3^5 appear even number of times, because 3^5 x 3 = 3^6.

Therefore we have

N x 3 = 2^10 x 3^5 x 3 = 2^10 x 3^6.

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step-by-step explanation:

he climbs 300 ft/hr at a constant rate. the peak is at 1,500 ft.

when going up, he reaches 600 ft in 2 hours. remember, that he is going in a constant rate. divide 600 with 300:

600/300 = 2

assuming that he takes the same route back (meaning that it's a 1500 ft route back, you can solve.

note also that you are solving when he gets back to 600 ft after he reaches the peak.

first, find the amount of hours it takes to get up the peak. divide the total amount (1500), with the ft/her (300) to get the amount of hours (x):

1500/300 = x

x = 5

it takes jamie 5 hours to get up. now, he is returning down, and reaches 600 (his current destination). remember that he climbs at a constant 300 ft/hr. divide 600 with 300:

600/300 = x

x = 2

now add the two numbers together.

x = 5 + 2 = 7

jamie takes 7 hours to go up the mountain and reach his current position when descending.

(total amount/constant rate) = (time).

remember that it is a constant rate, so in total, when he gets back down the mountain, it will take the same amount of time as when he goes up. the total amount for the round trip is: 5 + 5 = 10.

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