Divisibility by Four
Date: 04/19/97 at 19:55:32 From: Marianne Unruh Subject: Number How many 6-digit numbers are divisible by 4 if we allow no repeated digits?
Date: 04/20/97 at 15:26:37 From: Doctor Steven Subject: Re: Number We have a 6-digit number with no digit repeated and we want to know if it's divisible by 4. We are only worried about the last two digits since the first four digits are multiplied by 100, which is divisible by 4. ? ? --- --- --- --- --- --- Well, we know that the last digit has to be even since no odd number is divisible by 4. So for the last digit we have 5 choices: 0, 2, 4, 6, or 8. If we pick a number that is divisible by 4, then we need another even number for the second digit. So we would have 3 choices for the ones digit (since we can't have two of the same number) and then 4 choices for the tens digit. Then we would pick any sequence of digits for the hundreds and higher digits. This gives us 12*1680 = 20160 numbers divisible by 4. If we pick an even number that is not divisible by 4, then we need an odd number for the tens place. So we have 2 choices for our ones digits and 5 choices for our tens digit. Then we would again pick any sequence to fill in the remaining four digits. So we would get 10*1680 = 16800 numbers divisible by 4 this way. Summing these numbers up we get that there are 36,960 6-digit numbers divisible by 4 if we don't use any number twice. -Doctor Steven, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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