The third number
Number Theory
Level
3
Write the whole numbers from 1 to 37 in a straight line, so that each number divides the sum of all of the numbers previously written. If the first number is 37 and the second number is 1 what is the the third number?
The answer is 2.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
Is x the last number in the list. Then, by the assumptions of the problem, x must divide the number (1 + 2 + 3 + ··· + 37) − x, which implies that x divides (1 + 2 + 3 + ··· + 37) = 703 = 37 × 19. As x 6= 37, since 37 is the first on the list, it follows that x = 19. On the other hand, the third number in the list, y, has split 37+1 = 38 = 2×19. As y 6= 19, because 19 is the last number in the list, it follows that y =2