Try this beautiful problem from Algebra based on Prime numbers.
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?
Algebra
Number theory
card number
Answer:$14$
AMC-8 (2006) Problem 25
Pre College Mathematics
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 ..........
Obtain this even number would be to add another even number to 44
Can you finish the problem........
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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