Hi radi,

The information in this prompt can help you to make some deductions about P and R. We're told that they're both positive integers and that (P^2)/40 = R. The restriction that P cannot equal R means that P and R CANNOT equal 40. That restriction actually has no bearing on this prompt.

To start, we can get rid of the fraction:

P^2 = 40(R)

P^2 = (P)(P), so each of those parentheses MUST contain the SAME prime factors.

40(R) = (2)(2)(2)(5)(R)

Since there are two Ps, for each "P" to contain the same primes, we'll need an EVEN number of 2s and an EVEN number of 5s.

So far, we have three 2s, so we'll need AT LEAST one more 2. We also have just one 5, so we'll need AT LEAST another 5. Both that "extra" 2 and "extra" 5 MUST be in the R....

IF...R = 10, then 40(R) = (40)(10) = (2)(2)(2)(2)(5)(5) and P^2 = (2)(2)(5) = 20

This all proves 2 things:

1) R MUST be a multiple of 10 (but cannot be 40)

2) At the minimum, P = 20

Using this information, we can now deal with the Roman Numerals. Which of the following MUST be INTEGERS....?

I. R/5

Since R must be a multiple of 10, we know that R/5 is an integer.

Roman Numeral I must be an integer.

II. R/[(2)(5)]

Again, since we know that R must be a multiple of 10, R/10 is an integer.

Roman Numeral I must be an integer.

III. R/[(3)(5)]

R/15 may or may not be integer, depending on the value of R.

IF....R = 10, then R/15 is NOT an integer

IF....R = 30, then R/15 IS an integer.

Roman Numeral III is NOT necessarily an integer.

Final Answer:

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Rich

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