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Among any four consecutive odd numbers, one is divisible by $ 3 $, one by $ 5 $, one by $ 7 $, and no higher consistent prime appears.
Question: A glaciologist uses a base-eight elevation sensor reading of $ 2735_8 $ to calculate ice thickness. What is the equivalent base-ten elevation?
Solution: Convert $ 2735_8 $ to base ten:
Question: A retired engineer challenges students with a number puzzle: What three-digit positive integer less than $ 150 $ is two more than a multiple of $ 7 $, five more than a multiple of $ 9 $, and leaves a remainder of $ 3 $ when divided by $ 5 $?
Let $ x = 9k + 5 $. Substitute into the second:
Then $ x = 9(5m + 2) + 5 = 45m + 18 + 5 = 45m + 23 $
So $ m = 7n $, then $ x = 45(7n) + 23 = 315n + 23 $
Find $ x < 150 $: try $ n = 0 $: $ x = 23 $,
$ n = 1 $: $ x = 338 > 150 $, too big.
Solution: The area of the inscribed circle is: