00:00Let A, B, and C be positive integers.
00:03Given that A minus B minus 4C divided by A plus B minus 6C equals 2/3,
00:07The question asks for the smallest possible sum of A, B, and C.
00:11Let's write it down right away, guys, when you watch it.
00:13A minus B minus 4 times C divided by A plus B minus 6 times C equals 2 divided by 3.
00:21I'm doing the cross-multiplication right away.
00:233 times A minus 3 times B minus 12 times C equals.
00:282 times A plus 2 times B minus 12 times C.
00:33-12Cs cancel out the division.
00:35If we move 2A to this side and -3B to that side, folks...
00:383A minus 2A equals 3B plus 2B.
00:46Our A is equal to 5B, folks.
00:49So our assistant is actually the sum of A plus B plus C.
00:54If we substitute 5B for A, we are asked for 6B plus C.
00:59Okay, I know that the denominator needs to be different from zero here, right?
01:04And what else do I know, guys?
01:07What are positive natural numbers A, B, and C, friends?
01:13Because it says "positive integers".
01:17I could also write it like this, and it wouldn't be wrong.
01:21And friends, what is this?
01:23A, B, C are elements.
01:27Positive integers.
01:29They mean the same thing right now.
01:32When we look at this, friends...
01:35Now, to find the sum of A plus B plus C...
01:38positive natural numbers
01:39I can't just give zero instead of A, B, or C, can I, guys?
01:42I can't even give you the information.
01:43From where?
01:44It has to be different from zero, so what does that mean for positive natural numbers?
01:47Different from zero.
01:48It goes on forever, counting 1, 2, 3.
01:51Positive integers
01:52different from zero.
01:54Again, saying 1, 2, 3
01:55positive integers.
01:57Anything that goes in the positive direction goes on forever.
01:58So now I'm going to say...
02:01A plus B minus 6 times C
02:03I'll say it's not equal to zero.
02:07Right, guys?
02:08This is important.
02:09Here's something we're going to look at now.
02:11Friends, A plus B minus 6 C
02:13must be different from zero
02:16Where did that come from?
02:17From the denominator.
02:17If we replace A with 5B, friends...
02:206B minus, we found it from over there, right?
02:236B minus 6C
02:25I'll say it's not equal to zero.
02:29Then 6B is not equal to 6C.
02:33If you throw it this way, right guys?
02:35It should be 6C.
02:36So, in conclusion, friends...
02:38B needs to be different from C.
02:42That's what it means here.
02:44It says if B is equal to C.
02:45He says this place could be zero.
02:47This is the place, friends.
02:49The denominator must be different from zero.
02:51What happens if the denominator is zero, guys?
02:52Our expression would be undefined.
02:53It's not like 2/3 or anything like that.
02:54After looking at this, friends
02:56With your permission, I will try to give some value here.
03:00For the smallest value of A plus B plus C
03:02What can we give, friends?
03:03To A, B, C
03:06different from zero.
03:071, 2, 3, right?
03:08It would be different from B and C.
03:09In my total
03:10When I substitute A plus B plus C
03:12Wasn't it 6B plus C?
03:13Yes.
03:14So what am I going to do then, guys?
03:15Smallest instead of B
03:16What should I give, friends?
03:18I'll give it a 1.
03:19What should I use instead of C in this situation, guys?
03:24Of course, I should give a 2.
03:26I can't give a 0, right guys?
03:27Because he said it was positive
03:28natural numbers and integers.
03:30I'll give B a 1 and C a 2, guys.
03:33What will our A be in this case?
03:34A was equal to 5 times B.
03:37That equals 5, folks.
03:38With complete respect.
03:39Then our smallest A is 5
03:41Our B is 1
03:42Our C will also be 2, folks.
03:45Its smallest value is...
03:46We find our total to be 8.
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