Magic Prime

Logic Level 3

A blind-folded magician gave 10 balls, marked with the digits 0 to 9 distinctly, to the five volunteers and told each of them to make up 2-digit numbers from the 2 balls independently.

Magician: With my psychic power, I know that only one of you have a composite number and the other four have primes.

The audience cried out with disbelief, for it was all true.

Magician: And of those four primes, if they add the 2 digits together, three of those sums will be equal while the only one out is not. That prime with that special unequal sum will be called a magic prime...

Then when the magician spoke of that magic prime, all audience applauded loudly in unison.

What is the magic prime?

The answer is 61.

1 solution

Romain Bouchard
Jun 1, 2018

Here is the list of all $2$ -digits prime (with different digits) and the sum of their digits.

Prime #	$13$	$31$	$23$	$41$	$43$	$61$	$17$	$53$	$71$	$19$	$37$	$73$	$29$	$47$	$83$	$67$	$59$	$79$	$97$	$89$
Digit Sum	$4$	$4$	$5$	$5$	$7$	$7$	$8$	$8$	$8$	$10$	$10$	$10$	$11$	$11$	$11$	$13$	$14$	$16$	$16$	$17$

Since we know $3$ of the primes made by the volunteers have the same digit sum and are made of different digits, it has to be $\{29,47,83\}$ .

The only digits left are $0,1,5,6$ and the only prime that we can make from them is the magic prime : $\boxed{61}$ .

And what about : 07, 43, 61, 89 and 25 ??

Mr. India - 2 years, 6 months ago

Magic Prime

The answer is 61.

1 solution

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