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Chapter 10: Problem 1

If a prime number, \(p,\) is squared and the result is added to the next prime number greater than \(p\) which of the following integers could be the resulting sum? Indicate all such integers. a. 3 b. 4 c. 7 d. 14 e. 58 f. 60 g. 65 h. 69

Short Answer

Expert verified
The integers which could be the resulting sum are: c. 7, e. 58, h. 69.

Step by step solution

01

Identify Prime Numbers and next prime

The prime numbers less than 10 are: 2, 3, 5, 7. The next prime numbers are: 3, 5, 7, 11. We start with the smallest prime numbers because their square will be the smallest and thus likely to correspond to the given options.
02

Calculation

Square each prime number and add the next prime number to the result. For example, the first prime number is 2. Squaring 2 gives 4 and adding the next prime number (3) will result in 7. Carry out this operation for all identified prime numbers.
03

Compare with Given Options

Verify which sums are in the provided options. In this case, the calculation for the prime number 2 gave the result 7, which matches option c. Continue with this method until all prime numbers less than 10 (or until a suitable stopping point) have been processed.
04

Identify the remaining option(s)

Prime numbers larger than 7 have to be taken into account, because squaring e.g. 11 and adding 13 we get 142, which is not in the options. It's worth noting that every prime number greater than 3 is odd, so the squared value will also be odd, and when adding the next prime number (also always odd, if greater than 2), the sum will be even. Meaning that option g. 65 is not possible as well.
05

Verification

The remaining options e. 58, f. 60 and h. 69 can be tested with prime numbers greater than 7. After testing, we find that option e. 58 equals to \(7^2 + 11 = 58 \) and option h. 69 equals to \(5^2+59=69\). Thus, the options e. 58 and h. 69 are the remaining solutions.

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