Miscellaneous Problems

Problem 5. Factor 111111 (six 1’s) into primes.

Problem 4 Given that there is a unique digit d {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} so that the nine-digit number 19700019d is prime, find it.

Problem 3. Four comrades are racing side by side down a dusty staircase. Frodo goes down two steps at a time, Gimli three, Legolas four, and Aragorn five. If the only steps with all four’s footprints are at the top and the bottom, how many steps have just one footprint?

Problem 2. All perfect numbers known to date are even, and it is a conjecture, still unsolved, that there are no odd perfect numbers. Prove that 496 is a perfect number by finding all its proper divisors and adding them.

Problem 1. An unsolved problem in Number Theory is the Goldbach Conjecture, which asserts that every even natural number greater than 6 can be written as the sum of distinct primes. Verify Goldbach’s conjecture from every even integer between 8 and 36, inclusive.