She let him write it. He did, in words first, then in the compact signs his coach had permitted. 7k ≡ 1 mod 5. He glanced up to see whether the symbols were allowed in a diner. The diner, indifferent, went on being a diner. A man laughed too loudly at something on his screen. The sisters argued, with love, about whether their mother had always been like that.
“Seven and five,” Emily said. “Seven is two more than five. In the world of fives, seven behaves like two. So two k leaves a remainder of one.”
Caleb’s pencil moved. He whispered, “2k ≡ 1,” and then stopped. “I have to multiply by the thing that undoes two. The inverse. Mod five, two times three is six, and six is one more than five, so three undoes two. Multiply both sides by three. k ≡ 3. So k is three more than a multiple of five.”
He said it in a rush and then looked startled that the rush had arrived somewhere. Denise’s eyebrows lifted, not in mockery. She had not known he could speak this quickly without being afraid of the speed.
“Write the substitution before you admire it,” Emily said, and there was, for the first time, the ghost of a smile at her mouth. “Admiring is how you drop a sign.”