"The Staircase Race" - Futility Closet

A puzzle from Henry Dudeney’s Modern Puzzles and How to Solve Them, 1926:

This is a rough sketch of the finish of a race up a staircase in which three men took part. Ackworth, who is leading, went up three risers at a time, as arranged; Barnden, the second man, went four risers at a time, and Croft, who is last, went five at a time.

Undoubtedly Ackworth wins. But the point is, How many risers are there in the stairs, counting the top landing as a riser?

I have only shown the top of the stairs. There may be scores, or hundreds, of risers below the line. It was not necessary to draw them, as I only wanted to show the finish. But it is possible to tell from the evidence the fewest possible risers in that staircase. Can you do it?

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If the staircase were such that each man would reach the top in a certain number of full leaps, without taking a reduced number at his last leap, the smallest possible number of risers would, of course, be 60 (that is, 3 × 4 × 5). But the sketch showed us that A. taking three risers at a leap, has one odd step at the end; B. taking four at a leap, will have three only at the end; and C. taking five at a leap, will have four only at the finish. Therefore, we have to find the smallest number that, when divided by 3, leaves a remainder 1, when divided by 4 leaves 3, and when divided by 5 leaves a remainder 4. This number is 19. So there were 19 risers in all, only four being left out in the sketch.

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An idler's miscellany of compendious amusements

“The Staircase Race”