A Cunning Mathematician Vs A Magician

Number Theory Level 3

A cunning mathematician goes to a circus and watches the show. Afterwards, he says to the magician how silly and obvious his tricks are. The magician responds only with: "Do you do tricks? If you do, I bet they're more obvious than mine." The mathematician responds with a question he had recently challenged himself with.

"The following statements about X are true:

divisible by 2
not a prime number
a multiple of 4
a multiple of 3
a multiple of 8
more than Y

The following statements about Y are true:

a factor of 144 and 36
is a square number
not a factor of X
smaller than X
is more than 4

If X is between 1 and 30 and Y is between 1 and 40, what is the product of X and Y?"

The magician ponders for a moment and then says he can't answer the question. What is the answer to the mathematician's question?

33 72 216 266 288 864 432

4 solutions

Dwaipayan Shikari
Oct 14, 2020

The first number came in my mind for $X$ is $24$ . It is the smallest number which is a multiple of $3,4$ and $8$ . Obviously it is not a prime number.And divisible by $2$ and within the range of $(1-40)$

Then a square number which is a factor of $144$ and $36$ is $9$ . There are no number except it . And smaller than X and greater than 4 So $X=24$

And $Y=9$

Magician Number $=24×9=216$

Fletcher Mattox
Oct 13, 2020

sometimes writing code is easier than thinking. :)

from math import sqrt
for x in range(1, 31):
    for y in range(1, 41):
        if x%24 == 0 and x>y and 144%y == 0 and 36%y == 0 and sqrt(y) == int(sqrt(y)) and x%y != 0 and y > 4:
            print(x*y)

Richard Desper
Oct 14, 2020

$X$ is a multiple of $3$ and a multiple of $8$ , thus it is a multiple of $24$ . $X$ is also between $1$ and $30$ . So $X = 24$ .

$Y$ is a "factor of $144$ and $36$ " which means it's a factor of $36$ (which itself divides $144$ ). $Y$ is square, so it must be $1,4,9$ , or $36$ . $Y$ is less than $X$ , which rules out $36.$ $Y$ is not a factor of $X$ , and that rules out $1$ and $4$ . Thus $Y=9$ .

$XY = 24*9 = 216.$

Raph White
Oct 13, 2020

Y = 9, and X = 24. Y = 9 because 9 is a square, goes into 144 and 36 is more than 4. X = 24 because it is not a prime number, it is a multiple of 4, 3 and 8, is divisible by 2, and is more than Y. 24 times 9 is difficult, so

9 x 12 x 2 = 108 x 2 = 216.

A Cunning Mathematician Vs A Magician

4 solutions

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