Brain Teasers
Threes, Fives and Sevens
Which is the smallest natural number that satisfies the following conditions:
(1) Its digits consist only of, and only of 3's, 5's and 7's.
(2) Its digital sum is divisible by 3, 5 and 7.
(3) The number itself is divisible by 3, 5 and 7.
(1) Its digits consist only of, and only of 3's, 5's and 7's.
(2) Its digital sum is divisible by 3, 5 and 7.
(3) The number itself is divisible by 3, 5 and 7.
Hint
It's easier than some of you might think.Answer
From condition 2, its digital sum is at least 105.From condition 3, it must end with 5, but its divisibility by 3 is guaranteed, so we only have to check 7.
From condition 1 and 2, it has an odd number of digits. 105/7=15, so it has more than 15, and at least 17 digits. We can have:
(1) 13 7's, 1 5, and 3 3's;
(2) 12 7's, 3 5's, and 2 3's;
(3) 11 7's, 5 5's, and 1 3.
In the second case, we verify that neither 33557777777777775 nor 33575777777777775 is divisible by 7, but 33577577777777775 is divisible by 7 (the quotient is 4796796825396825.)
In the first case, the only number they make that is less than 33577577777777775 is 33377777777777775, which is not divisible by 7.
In the third case, they can't make a number less than 33577577777777775.
Therefore, the number we're looking for is 33577577777777775.
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Comments
I like some of these, but this one is just plain stupid.
that was so difficult for me.
hard and boring
but mostly stupid
oohh tough one, had to get a sheet of paper and a pencil to solve
At the time that I rate this teaser, it has a difficulty rating of 2.95. How can a difficult teaser be stupid?
Then why was 2224 rated low in "fun"?
"its divisibility by 3 is guaranteed"?? your familarity with field has led you to skip past explanation, or even naming specific area of study which would declare and explain same.
it scored poorly because question is more task than teaser. It is difference between yeah and aha.
it scored poorly because question is more task than teaser. It is difference between yeah and aha.
Actually, I thought it was kind of fun.
It would have been tedious if I'd had to go much past the fourth possible candidate to find the solution. It has the element of a teaser because you need to first figure the smallest possible number it could be and then work up from there. It takes some elementary deduction and didn't involve too much effort once the requirements were established.
It would have been tedious if I'd had to go much past the fourth possible candidate to find the solution. It has the element of a teaser because you need to first figure the smallest possible number it could be and then work up from there. It takes some elementary deduction and didn't involve too much effort once the requirements were established.
Thanks for your puzzle.
I managed to solve it with glpsol (a free linear and mixed integer programming solver, available for most platforms). I uploaded the model file that I used to pastebin for anyone who might be interested to play with it.
http://pastebin.com/7D04GYJn
I managed to solve it with glpsol (a free linear and mixed integer programming solver, available for most platforms). I uploaded the model file that I used to pastebin for anyone who might be interested to play with it.
http://pastebin.com/7D04GYJn
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