A Nice Algebra Problem | Math Olympiad a+b=?

本站添加: 2026-05-10

[00:00:00]Hello everyone. Welcome to learn to solve this very nice Diophantine equation. A plus B over 2 AB is equal to 1 over 14. Our job is to find all possible values of A plus B. Such that A and B are integers. So let's start.

[00:00:18]Since uh A times B is in the denominator, so A cannot be equal to zero and B cannot be equal to zero.

[00:00:32]Now we multiply both sides of this equation by two.

[00:00:35]The equation is A plus B divided by 2 AB equal to 1 over 14. And we multiply both sides by two.

[00:00:52]So this two will be canceled out with this two and two one time two seven times.

[00:00:58]So the equation will become A plus B divided by A times B equal to 1 over 7.

[00:01:14]Now to further simplify, we use cross multiplication. We multiply this seven by A plus B and we multiply this AB by this one.

[00:01:23]So this will become seven times uh A plus B equal to AB times 1 AB.

[00:01:35]Distribute this seven. Seven times A will become 7A plus seven times B 7B equal to A times B. Move these two terms to the right hand side then this will become A times B minus seven times A minus seven times B equal to zero.

[00:02:00]From these two terms AB minus 7A, we can factor out A.

[00:02:07]In bracket left B minus seven.

[00:02:12]This negative 7 * B remains same equal to zero.

[00:02:22]Now because we have B minus seven as a factor, so to get the same factor here, we add 49 to both sides. At left hand side, we add 49.

[00:02:37]And at right hand side, we also add 49.

[00:02:42]So this left hand side will become A times B minus seven.

[00:02:49]From these two terms, we can factor out negative seven. In bracket left B minus seven.

[00:02:57]Equal to zero plus 49 is 49.

[00:03:03]Now this B minus seven is a common factor, so we factor out this B minus seven.

[00:03:10]And in bracket left A minus seven.

[00:03:17]Equal to 49.

[00:03:20]We can rewrite this expression at the left hand side as A minus seven times B minus seven.

[00:03:31]Equal to 49.

[00:03:34]Now we have a product of two factors at left hand side.

[00:03:39]And because both A and B are integer, it means that this A minus seven A minus seven is also an integer.

[00:03:56]And this uh B minus seven will also be an integer.

[00:04:07]And uh integer factors of this 49, integer factors of this 49 are one time 49 seven time seven 49 times one negative one times negative 49 and negative seven times negative seven and negative 49 times negative one.

[00:04:42]It means that we have six cases. This is case one.

[00:04:46]This is case two. This is case three.

[00:04:49]This is case four. And this is case five. And this is case six.

[00:04:55]First, we solve this case one, one time 49.

[00:05:01]So, in case number one in case number one we have this expression A minus seven A minus seven will be equal to this one.

[00:05:16]And the second factor B minus seven will be equal to this second factor 49.

[00:05:25]49 If we add seven to both sides, this implies that A is equal to eight.

[00:05:33]And if we add seven to both sides of this equation this implies that B is equal to 56.

[00:05:42]And A plus B will be equal to 8 plus 56.

[00:05:52]A plus B will be equal to 64.

[00:05:57]4.

[00:05:58]So, from this case we get the first solution of the equation A plus B 64.

[00:06:07]Now, we solve this second case as 7 * 7.

[00:06:12]In case number two, we write this factor A minus 7 A minus 7 equal to 7.

[00:06:23]And the second factor B minus 7 equal to 7.

[00:06:35]If we add 7 to both sides of this equation, this implies that A is equal to 14.

[00:06:43]And if we add 7 to both sides of this equation, this implies that B is equal to 14. And uh A plus B will be equal to 14 plus uh 14.

[00:07:00]So, A plus B will be equal to 28.

[00:07:05]So, from this second case we get the second solution A plus B equal to 28.

[00:07:13]Now, we solve this third case number three 49 * 1.

[00:07:19]In case number three, we write this factor A minus 7 A minus 7 equal to 49.

[00:07:31]And the second factor B minus 7 equal to 1.

[00:07:41]If we add 7 to both sides of this equation, this implies that a is equal to 56.

[00:07:49]And if we add seven to both sides of this equation, this implies that b is equal to eight.

[00:07:55]So, a plus b will be equal to 56 plus eight.

[00:08:03]And a plus b will be equal to 64.

[00:08:09]So, from the third case, we get a same solution as we get in case one, 64.

[00:08:17]Now, we move on to case number four. In case number four, in case four, we have -1 * -49.

[00:08:34]In case number four, we write this first factor a - 7 = -1.

[00:08:41]And this second factor, b - 7 = -49.

[00:08:49]From this equation, if we add seven to both sides, this implies that a is equal to six.

[00:08:57]And from this equation, if we add seven to both sides, this implies that b is equal to 42.

[00:09:08]So, a plus b will be equal to six minus 42.

[00:09:16]And a plus b will be equal to 36.

[00:09:23]So, from this case, we get another solution, a plus b equal to -36.

[00:09:31]Now, we move on to case number five. In case number five, we have In case five, we have this factor -7 * -7 in case number In case number five, we write this factor a -7 equal to -7 and we write the second factor b -7 equal to -7.

[00:10:09]If we add seven to both sides of this equation, this implies that a equal to zero.

[00:10:17]And if we add seven to both sides of this equation, this implies that b is equal to zero.

[00:10:26]And since a and b are in the denominators, so a cannot be equal to zero and b cannot be equal to zero.

[00:10:39]This will be rejected and this will be rejected. From this case, we get no solution.

[00:10:48]Now, we move on to case number six. In case six, we have this factor -49 * -1.

[00:11:05]In case number six, we write this first factor a -7 equal to -49 and the second factor b -7 equal to -1.

[00:11:22]If we add seven to both sides of this equation, this implies that a is equal to -42. 42.

[00:11:32]And if we add seven to both sides of this equation, this implies that B is equal to six.

[00:11:41]It means that A plus B will be equal to negative 42 plus six.

[00:11:50]And this implies that A plus B will be equal to negative 36.

[00:11:58]So, from this case, we get another solution, negative 36.

[00:12:06]So, the final set of solutions of A plus B is equal to The first is 64 and second is 28.

[00:12:28]And the third one is negative 36.

[00:12:32]So, the set of solutions of A plus B will be equal to 64, 28, and negative 36.

[00:12:47]This is the final answer of this problem.