Try this beautiful problem from Algebra based on Prime numbers.

Algebra based on Number theory – AMC-8, 2009 – Problem 23


Barry wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table, as shown. The sums of the two numbers on each of the three cards are equal. The three numbers on the hidden sides are prime numbers. What is the average of the hidden prime numbers?

  • $14$
  • $12$
  • $16$

Key Concepts


Algebra

Number theory

card number

Check the Answer


But try the problem first…

Answer:$14$

Source
Suggested Reading

AMC-8 (2006) Problem 25

Pre College Mathematics

Try with Hints


First hint

Notice that 44 and 38 are both even, while 59 is odd. If any odd prime is added to 59, an even number will be obtained.

Can you now finish the problem ……….

Second Hint

Obtain this even number would be to add another even number to 44

Can you finish the problem……..

Final Step

Notice that 44 and 38 are both even, while 59 is odd. If any odd prime is added to 59, an even number will be obtained. However, the only way to obtain this even number would be to add another even number to 44 , and a different one to 38. Since there is only one even prime ( 2 ), the middle card’s hidden number cannot be an odd prime, and so must be even. Therefore, the middle card’s hidden number must be 2, so the constant sum is 59+2=61. Thus, the first card’s hidden number is 61-44=17, and the last card’s hidden number is 61-38=23

Since the sum of the hidden primes is 2+17+23=42, the average of the primes is \(\frac{42}{3}=14\)

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