Rive

Rive's given name was Bram, but he had been called Rive since he was twelve, when a mason asked him for a block that would tile a courtyard exactly, and Bram walked the whole quarry, tapping blocks, until he found one that would rive into the mason's precise number of rows with nothing wasted. "That boy can rive a number to order," the mason said, and the name held.

He came to think of every number as a block waiting to be tested. To find a number's factors — the numbers it splits into equally — you simply tried to rive it: does it break into rows of two? rows of three? rows of four? Each clean split named a pair of factors. Twelve broke at two (giving 2 and 6), at three (giving 3 and 4), and there was no need to look past its own square-root, because after that the pairs only repeat in the other order.

And the stubborn blocks — the ones that broke at nothing but one-times-themselves — those he learned to call primes. Two, three, five, seven, eleven, thirteen: the un-riveable ones. Not broken. Whole.

An academy quarrier came for building-stone and found Rive sorting a season's blocks into "splits cleanly" and "won't split," and he stopped, because that was exactly the lesson he could never get children to feel.

"You are factoring," he said. "And your stubborn stones are the prime numbers. We teach that every number is either built from primes or is a prime itself, and children think it is a rule from nowhere. You have it in your hands."

Rive left the quarry the next spring. He has taught factors and primes for eight years, and he keeps, on his classroom floor, a set of small wooden cubes for riving.

Rive begins each year with twelve cubes. "Rive them into equal rows," he says, and the children discover, delighted, that twelve is generous — two sixes, three fours, four threes, six twos. He writes each split as a pair: 2×6, 3×4, and shows that these pairs are the factors of twelve: 1, 2, 3, 4, 6, 12. "A factor," he says, "is any number that rives it cleanly. Twelve has six of them. Twelve is an easy stone."

Then he hands them seven cubes and says, "Now rive these." The children try. Two rows leave one over. Three rows leave one over. They cannot do it. And Rive smiles and says, "You're not failing. Seven is a prime. It won't rive into equal rows except one row of seven. Its only factors are one and itself. It is not a broken number — it is a building number." Then he shows them the point: he rives thirty into 2×15, then rives the 15 into 3×5, and holds up the primes at the bottom — 2, 3, 5 — and says, "Every number, if you keep riving, comes down to a handful of primes. The primes are the blocks. All the others are walls built from them."