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# Number Theory

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Find the largest positive integer \$n\$ such that \$n^3\$ divides 15.

Jul 22, 2024

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We're converting the power of a microwave from Watts (W) to a new unit, Zaps (z). We know the conversion rates for both Watts and Zaps to SI units (kg, m, s, min).

Watts to SI:

1 W = 1 kg * m^2 / s^3

Zaps to SI:

1 z = 1 kg * m^2 / min^3

We want to find the power of a 900 W microwave in Zaps. To do this, we can write an equality where the power is the same but expressed in different units:

900 W = P z

Now we can manipulate the equation to solve for P (power in Zaps). We can do this by introducing a conversion factor that equates Watts and Zaps.

This factor will cancel out the desired units (kg and m^2) and leave us with a factor that relates Watts and Zaps through time units (seconds and minutes).

a. We know from the definitions of Watts and Zaps that:

- 1 W / (1 kg * m^2 / s^3) = 1 z / (1 kg * m^2 / min^3)

b. This simplifies to:

- 1 W * (min^3 / s^3) = 1 z

c. This conversion factor is equal to 1 because we're converting between equivalent units that express the same fundamental quantities (mass, length, time) but in different time scales (seconds vs minutes).

Apply the conversion factor to the original equation:

900 W * (min^3 / s^3) = P z

Since the conversion factor is 1 (as derived previously), we have:

900 W = P z

Therefore, the power of the 900 W microwave in Zaps is also 900 Zaps. However, to express the answer in scientific notation, we should recognize that 900 can be written as 9.00 x 10^2.

Answer: The power of the 900 W microwave in Zaps is 9.00 x 10^2 Zaps.

Jul 24, 2024