Math Musings

Chika Ofili, a Nigerian boy, 12-year-old living in London just discovered a new Formula for divisibility by 7 in maths.  Multiply the last digit by 5 and add it to the remaining numbers.  For Example 532 53+2 * 5 =63 which is a multiple of 7, So 532 is divisible by 7. Chika Ofili has been presented with a special Recognition Award for making a new discovery in mathematics Here is a simple Proof:- If x+5b=0 mod 7 then multiplying by 10 we get 10x+50b=0 mod 7.  But 50 is just 1 mod 7 so we get 10x+b=0 mod 7 as an equivalent statement.  

Four runners compete in a race: Annie, Becca, Carlos, and Dante. After some confusion at the finish line, it's unclear what the final finishing order was, but the following information is known:  1. Dante finished before Annie.  2. Becca wasn't third.  3. There were two runners between Annie and Carlos.  Who won the race?  Puzzle Answer