Question

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(1) In how many ways can the digits 1 to 9 be arranged in three rows, such that the number in the first row is half of the number in the second row and a third of the number in the third row, while using each digit exactly once? [Solved by @KingGeorge]

(2) What is the minimum possible LCM of 10 natural numbers whose sum is 45? [Solved by @KingGeorge, @dumbcow @callisto]

**NOTE**: Please don't post the solution here, instead use this thread for clarification of problem statement discussing strategy and checking the answer. The reason for this is once a solution is posted it act as a spoiler for others. At-least wait for a day before posting a solution. And If you are really confidant about your solution send my via private message.

Good luck!

Regards,
Fool/FFM!

Answer 1

Thanks for the problems, mate. Just wondering, for (1), do we have to use all 1-9 digits or enumerate the possible solutions for 3 rows?

Answer 2

You are welcome :)

Yes, we have to use all 9 digits at-least once.

Answer 3

Thanks for the reply @FoolForMath :-)

Answer 4

The first problem reminds me of some of the questions badreferences recently posted.

Answer 5

Isn't there a way to mark a question to be good? :D

Answer 6

There was but the admins don't like good question anymore.

Answer 7

lol "welcome" i wouldnt wanna be welcome to those :P

Answer 8

Is 0 included as a natural number?

Answer 9

Nope.

Answer 10

to get a three digit number and not a four digit number in the third row we can make the restriction that the top left number is less or equal to 3

Answer 11

anyone want to confirm that? im not 100%

Answer 12

Your thinking is correct. However, we can actually do a little better than three. If it were 3, the other two digits would have to be 2 and 1, and neither 321 or 312 work. Thus, we can even restrict it to less thatn or equal to two.

Answer 13

nice

Answer 14

Actually, now that I think about it, I think what I just said was incorrect. As long as the first number is less than 334 we should be fine. So we can almost restrict it to less than three. But there's a few numbers in there we're missing.

Answer 15

For the first question, does it mean that we can only use the digits 1-9 only once to arrange 3 rows of number? ( and the numbers arranged are of different digits, say, 123 456 789, but not 123 456 7789?)

Answer 16

Only once.