Ioana picks out two numbers, a and b, from the set 1, 2, 3, all the way up to 26. The product a, b is equal to the sum of the remaining 24 numbers. What's the value of the absolute difference of a and b? So let's call these numbers, let's find out the sum of the remaining 24 numbers. Well that's just the sum of everything minus a minus b. So the sum of everything from 1 to 26 is just 26 times 27 over 2, and then we subtract out a and b, and now this is supposed to be equal to the product of a and b. Okay so now let's just simplify everything and move this a and b to the other side to get ab plus a plus b is equal to 351. Well this looks very close to factoring, it just looks like it's missing a constant term at the end here, so let's just add it in. ab plus a plus b plus 1, so that this side factors as a plus 1 times b plus 1, and then it's equal to 352. So now we need to find some way to multiply two numbers to get 352, so let's first factorize 352 as 2 to the 5th times 11. And now we need to come up with some way of having both of these numbers be between 1 and 26, and having it be 16 times 22 does the trick. One way to come up with this is to say that 11 has to be in one of the factors, and if we just do 11 times 32, 32 is too big. So let's just steal a 2 and put it in here, and we get 16 times 22. And the thing is if we take another factor of 2 and put it in here, we get 8 times 44, and 44 is just too big. Okay so that means a and b have to be 16 and 22 in some order, so the absolute value of their difference is 6. So we get 6.

number selection

product and sum

factoring

equation transformation

absolute difference