Last week, in Number Sense 036, Awkward Goat challenged the Marching Goat Society to disprove his conjecture that every time they got into formation, their rows and their columns were awkward numbers, or the product of awkward numbers. The Marching Goats couldn't do it with only ten goats, but they vowed to return this week, with many more goats, to prove Awkward Goat wrong.
A lot more goats came to march in formation this week. They had all heard about Awkward Goat's challenge. Billy Goat Gruff stepped forward as spokesgoat for the crowd.
“We have a lot more goats this week,” said Billy Goat Gruff. “I'm sure we can get into a formation that doesn't involve awkward numbers in either our rows or our columns.”
“Let's suppose you can,” replied Awkward Goat.
“I like the sound of that already,” said Billy Goat Gruff.
“But you couldn't do it with ten goats last week,” Awkward Goat reminded him.
“A ten goat formation has two columns and five rows, and both two and five are awkward numbers, or, as I like to call them, primary numbers,” said Awkward Goat.
“That's why I brought more goats this week,” replied Billy Goat Gruff.
“As I said, let's suppose you can do it,” said Awkward Goat. “Let's suppose you can make a column of goats that is not a primary number and is not a product of primary numbers.”
“Just one column?” asked Billy Goat Gruff suspiciously. “We march in formation, you know, not single file.”
“I know that,” replied Awkward Goat. “I just thought that if you could find a number for a single column, you could just use the same number for the rows. You don't have to find two numbers to prove me wrong.”
“That makes sense,” said Billy Goat Gruff, “but I think you are giving in too easily.”
Billy Goat Gruff turned to the assembled goats. “Marching Society!! Single Column!! Form UP!!!”
“This could take a while,” said Billy Goat Gruff. “That's a lot of goats. I'm not sure I can count that high.”
“Maybe we don't have to,” said Awkward Goat. “We know that numbers up to ten are either primary or products of primary numbers, so the number we are looking for is bigger than ten.”
“And there might be many such numbers, but if we go in order, starting with eleven, sooner or later we will find the first number that is not primary nor a product of primary numbers. If we keep going after that, we may find more, but that first one will be the smallest such number.”
“Well,” sighed Billy Goat Gruff, “we may as well get started.”
“Wait just a bit,” said Awkward Goat. “Let's suppose that you've got the number standing there right now.”
Billy Goat Gruff perked up, “Then I win! I'm right, and you are wrong!”
“Not quite,” replied Awkward Goat. “You agree that number cannot be a primary number, right?”
“An AWKWARD number, right.”
“Ok, an awkward number, whatever. But if it's not an awkward number, then you can make a marching formation out of that many goats, right?”
“Of course I can. MARCHING SOCIETY!! Rows AND columns!! Form UP!!!” bellowed Billy Goat Gruff.
“There you are,” said Billy Goat Gruff. “A nice, neat marching formation.”
“And you agree that in this formation, the rows and columns are both awkward numbers or the product of awkward numbers?”
“Why should I agree with that?” asked Billy Goat Gruff, suddenly suspicious again.
“Because you agreed that the total number of goats standing there was the smallest number that wasn't a primary number or the product of primary numbers. Since the number of goats in each row and the number of goats in each columns are obviously smaller than the number of goats in the whole formation, they are both smaller than the smallest number that is neither a primary number nor the product of primary numbers.”
“I suppose so,” said Billy Goat Gruff.
“Just suppose so?”
“All right, it IS so. The rows and columns are awkward numbers,” conceded Billy Goat Gruff.
“Primary numbers,” Awkward Goat reminded him, “or the product of primary numbers. And how do we figure out the number of goats in formation if we know the number of rows and the number of columns?”
“You're enjoying this, aren't you,” grumbled Billy Goat Gruff. “You multiply the number of rows by the number of columns. The total number of goats is the product.”
“The product of primary numbers,” finished Awkward Goat.
“But we supposed that we could find that smallest non-primary non-product of primary number. We supposed that we DID find it!”
“When we supposed that,” answered Awkward Goat, “we were wrong. It turns out we just found another number that was a product of primary numbers. The number we were looking for doesn't exist.”
“But what if we'd just gone a bit further? What if I'd brought even more goats?”
“It wouldn't have made a difference. If those numbers are out there at all, there has to be a smallest one. And the smallest one turns out not to be that kind of number after all.”
Have fun in the comments.