Rive

"Some numbers split. Some don't," his aunt said, sighting a chisel. "The ones that won't — they're the stubborn ones. They're also the ones everything else is built from, though it took me years to see it."

That sorting — splits cleanly versus won't split — is exactly what the academy calls finding factors and identifying primes. A factor is any number that rives another into equal whole rows with nothing left; a prime is a number greater than one whose only factors are one and itself.

Rive's given name was Bram, but he was called Rive from the day a mason asked for a block that would tile a courtyard exactly and he walked the whole quarry tapping stone until he found one that rived into the mason's precise number of rows with nothing wasted. He came to treat every number as a block waiting to be tested: does it break into rows of two? three? four? Each clean split names a pair of factors — twelve breaks at two (giving 2 and 6), at three (giving 3 and 4). And here Rive teaches the shortcut the quarry taught him: you never have to test past a number's square root. "Once I reach the row-size whose square passes the block," he tells students, "every new split I'd find is just a pair I already found, flipped. Four-times-three is three-times-four wearing the other coat. So I stop at the square root and I have them all." It halves the labour and misses nothing — a symmetry, not a guess.

The stubborn blocks — two, three, five, seven, eleven, thirteen — he learned to call primes. "Not broken," he insists. "Whole. Un-riveable." And then the deep claim, the one the academy calls the fundamental fact of arithmetic: every number is either a prime or is built, in exactly one way, from a product of primes. Rive shows it by riving thirty into 2×15, then riving the 15 into 3×5, and holding up the bottom row — 2, 3, 5 — the primes from which thirty is made. "Keep riving any number," he says, "and it comes down to a handful of primes, always the same handful, in only one way. The primes are the blocks. Every other number is a wall built from them."

An academy quarrier came for building-stone and found Rive sorting a season's blocks into "splits cleanly" and "won't split." "You are factoring," he said, "and your stubborn stones are the primes. We teach that every number is built from primes, and children think it a rule from nowhere. You have it in your hands." Rive left the quarry the next spring and has taught factors and primes for eight years, keeping a set of small wooden cubes on his classroom floor for riving.

He begins each year with twelve cubes. "Rive them into equal rows." The children find twelve generous — two sixes, three fours, four threes, six twos — and Rive writes the factor list: 1, 2, 3, 4, 6, 12. Then he hands them seven cubes: "Now rive these." They try; two rows leave one over, three rows leave one over; they cannot. "You're not failing," Rive smiles. "Seven is a prime. Its only factors are one and itself — a building number, not a broken one." Then he rives thirty down to 2, 3, 5 and lets them see the blocks at the bottom.