Brain Teasers
3 to 40
I am thinking of an even number. The ratio of the sum of its digits to the product of its digits is 3 to 40 . The number is divisible by (40-3) .
What is the smallest number that meets these requirements??
What is the smallest number that meets these requirements??
Hint
Sum of the digits = 18Product of the digits = 240
Answer
5328 is the number.3/40 = 18/240
S.O.D. = 18; P.O.D. = 240
5328 / 37 = 144
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Comments
I do not understand the (40-3). It looks like an underscore, teh way it comes up on my screen. Why not say the difference of the ratios or just plain 37? Can you explain where you got the 5238? I know it's the answer but how?
Thanks Jimbo. The reason I wrote 40-3 instead of 37 was to keep with the 40 and 3 "theme" of the puzzle.
First, notice that the ratio of 3:40 is equivalent to each of the ratios 6:80 , 9:120, 12:160, 15:200, and 18:240 . The only one of these in which the actual sum of the digits is the first number in the ratio and the actual product of the digits is the second number in the ratio is 18:240 . The digits MUST be 2, 3, 5, and 8 . Now, there are 24 numbers that have the digits 2, 3, 5, and 8 . But since we know that our number is EVEN, then there are just twelve possibilities. Of those 12, only one is divisible by 40-3,or 37. And that is the number... 5328
First, notice that the ratio of 3:40 is equivalent to each of the ratios 6:80 , 9:120, 12:160, 15:200, and 18:240 . The only one of these in which the actual sum of the digits is the first number in the ratio and the actual product of the digits is the second number in the ratio is 18:240 . The digits MUST be 2, 3, 5, and 8 . Now, there are 24 numbers that have the digits 2, 3, 5, and 8 . But since we know that our number is EVEN, then there are just twelve possibilities. Of those 12, only one is divisible by 40-3,or 37. And that is the number... 5328
I too was confused with the 40-3 part... I understood it to be just a further explanation of the ratio... so when I figured it out I came up with the twelve possibilities and nowhere else to go... maybe the wording can be altered for greater understanding? Like the answer is divisible by the difference between 40 and 3.
FANTASTIC!
There are other solutions in addition to 5328 (all keeping within the 18:240 ratio). For instance: 12654, 14652, 52614 and 54612.
delicious
Add 41625 to the list.
Not a good teaser since as far as I know the only way to tackle a problem like this is a brute force search. You can apply some heuristics to prune the solution space to search, but you can't solve this mathematically.
If there is a way to solve this without a brute force search, that should be part of the explanation in the answer.
Not a good teaser since as far as I know the only way to tackle a problem like this is a brute force search. You can apply some heuristics to prune the solution space to search, but you can't solve this mathematically.
If there is a way to solve this without a brute force search, that should be part of the explanation in the answer.
OK, I guess I should have read all the comments to see that the mathematical method of solution was there instead of in the teaser answer.
I don't understand what you mean by the sentence
"The only one of these in which the actual sum of the digits is the first number in the ratio and the actual product of the digits is the second number in the ratio is 18:240."
The actual sum of what digits? Do you mean to say the only one of these where there is a combination of digits where the sum is the first number and the product is the second number?
Obviously, the digits don't HAVE to be 2, 3, 5 and 8 since 1, 2, 4, 5, 6 also work.
There is a pretty big hole in your explanation for how you selected 18:240 from the multiples of the ratio. 4255111 also works and has a 15:200 ratio.
I don't understand what you mean by the sentence
"The only one of these in which the actual sum of the digits is the first number in the ratio and the actual product of the digits is the second number in the ratio is 18:240."
The actual sum of what digits? Do you mean to say the only one of these where there is a combination of digits where the sum is the first number and the product is the second number?
Obviously, the digits don't HAVE to be 2, 3, 5 and 8 since 1, 2, 4, 5, 6 also work.
There is a pretty big hole in your explanation for how you selected 18:240 from the multiples of the ratio. 4255111 also works and has a 15:200 ratio.
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