CAT Exam

Write the prime number between 200 and 300 which has the smallest digital root , as the sum of positive real numbers so as to maximize their product.

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THE PRIME NUMBER BETWEEN 200 AND 300 WITH LOWEST DIGITAL ROOT IS 271.
note that,

to maximize the product, all of the numbers should be equal.

we seek the maximum value of y = (271/x)^x, where x is a positive integer.
ln y = x · ln(271/x) = x (ln 271 − ln x)
(1/y) y' = ln 271 − ln x − 1
y' = (271/x)x (ln 271 − ln x − 1)

Setting y' = 0, x = 271/e is approximately equal to 99.7, which is clearly a maximum.
Now we need only try both 100 and 99 to confirm that 100 is the maximum value for y, when x is an integer.

Therefore the maximum product occurs when 271 = 2.71 + 2.71 + ... + 2.71. (100 equal terms.)
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