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2, 3, 5, 7, ... The set of prime numbers is the alphabet of mathematics that allows us to communicate across the universe.

Consider a digital clock. The first time after midnight when the three digits are all prime numbers is 2:22 am. What is the **last time before 10:00 am** when the three digits on the clock are all prime numbers?

**Details and assumptions**

If your answer is 1:23 am, type it in as 123.

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John and Amy have the mathematical conversation above. How old is Amy?

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The sum of three primes is 100. One of them exceeds the other by 36. Find the largest prime.

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\[345!+2 , 345!+3 , 345!+4 , \ldots , 345!+345\]

Find the number of primes among the numbers above.

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If a prime \(p\) can be expressed as the difference of square of two positive integers, then find the remainder, when \(p^{2}+138\) is divided by \(4\).

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