2016 AMC 8 Problems/Problem 24
The digits , , , , and are each used once to write a five-digit number . The three-digit number is divisible by , the three-digit number is divisible by , and the three-digit number is divisible by . What is ?
Solution 1 (Modular Arithmetic)
We see that since is divisible by , must equal either or , but it cannot equal , so . We notice that since must be even, must be either or . However, when , we see that , which cannot happen because and are already used up; so . This gives , meaning . Now, we see that could be either or , but is not divisible by , but is. This means that and .
We know that out of is divisible by . Therefore is obviously 5 because is divisible by 5. So we now have as our number. Next, let's move on to the second piece of information that was given to us. is divisible by 3. So, according to the divisibility by 3 rule, the sum of has to be a multiple of 3. The only 2 big enough are 9 and 12 and since 5 is already given. The possible sums of are 4 and 7. So, the possible values for are 1,3,4,3 and the possible values of are 3,1,3,4. So, using this we can move on to the fact that is divisible by 4. So, using that we know that has to be even so 4 is the only possible value for . Using that we also know that 3 is the only possible value for 3. So, we have = so the possible values are 1 and 2 for and . Using the divisibility rule of 4 we know that has to be divisible by 4. So, either 14 or 24 are the possibilities, and 24 is divisible by 4. So the only value left for is 1. .
Solution 3 (Divisibility Rules)
We know that is divisible by , so would be either or . However, is not a choice, so . Also, is divisible by , so this means that is , , , or . If , then has to be or ( is divisible by ), but both are taken. So, . must equal or , but because , . This leaves
Solution 3 (Lucky and Fast)
We can simply try each of the answer choice, and we will see which one works. Trying , if is divisible by , must be divisible by four. Therefore, can only be , , or . However, since is divisible by , , so cannot be . When , , the last requirement cannot be satisfied because , and is not divisible by . However, when , , the last requirement can be satisfied. Hence, we can see that when , there is one way to satisfy all three requirements, leading to a conclusion that is .
Video Solution (CREATIVE THINKING + ANALYSIS!!!)
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