Question 1147094: Numbers less than 4000 are formed from the digits 1, 3, 5, 8 and 9, without repetition.How many of them are divisible by 3?
Found 2 solutions by Alan3354, greenestamps:
Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website! Numbers less than 4000 are formed from the digits 1, 3, 5, 8 and 9, without repetition.How many of them are divisible by 3?
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Step 1, find the numbers less than 4000.
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Step 2, use the fact that if the sum of the digits is a multiple of 3 it's divisible by 3.
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There's no algebraic way to do step 1.
I don't need the practice. You might.
Answer by greenestamps(13209)  (Show Source):
You can put this solution on YOUR website!
You can get some practice in some good mathematical techniques by working through this problem.
We are concerned with divisibility by 3; so we can do the analysis in modular arithmetic, mod 3. The digits we have, and their values mod 3, are
digit mod 3
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1 1
3 0
5 2
8 2
9 0
Since a number is divisible 3 if and only if the sum of its digits is divisible by 3, we want combinations of the values of the given digits, mod 3, that are divisible by 3.
(1) 1-digit numbers:
The single digit has to be equal to 0 mod 3.
Those digits are 3 and 9.
It's a bit absurd to ask the question here; but it follows a fixed procedure: how many 1-digit numbers can you make using either of those digits?
(2) 2-digit numbers:
There are two ways to get a combination of 2 digits that is equal to 0 mod 3: 0 and 0; or 1 and 2.
The 2-digit combinations with the given digits are 3 and 9; or 1 and 5, or 1 and 8.
How many 2-digit numbers can you make using any of those combinations of digits?
(3) 3-digit numbers:
There is only one way to get a combination of 3 digits that is equal to 0 mod 3: 0, 1, and 2.
The 3-digit combinations with the given digits are 3-1-5, 3-1-8, 9-1-5, and 9-1-8.
How many 3-digit numbers can you make using any of those combinations of digits?
(4) 4-digit numbers:
Note that the sum of the mod 3 values of all five of the given digits is 2 mod 3. That means the digit that must be left out of a combination of 4 digits is one of those whose value mod 2 is 2; those digits are 5 and 8.
So the 4-digit combinations of digits are 1-3-5-9 and 1-3-8-9.
How many 4-digit numbers can you make using any of those combinations of digits? Note in this case that the problem requires that the 4-digit number has to be less than 4000, so the leading digit must be either the 1 or the 3.
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