Thursday, November 12, 2015

POTW #11 - Can You Digit?

Please see the answer to POTW 10 below. Do you see where and why you ontained a different result than 9:4? POTW #11 follows after the answer below.



POTW 11

21 comments:

  1. Since I had to make the largest positive difference between 5A4 and 37B, I knew I had to make one of them as biggest as possible, and the other smallest as possible. Since number 5A4 started with a 5, and number 37B started with a 3, I knew which number was set to be the largest.

    5A4 = 584 37B = 372
    A = 6 B = 2
    Since all numbers divisible by 4 have to be even and 5A4 had to be the largest, I made it the largest single digit, even number; 8. I did the same with the number 37B, but the opposite. I knew the number had to be divisible by 3, and the smallest as possible. That had to be 2 since using numbers 0 and 1 gave decimals.

    584 - 372 = 212 is the largest difference between the two digits

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  2. The largest number 5A4 can be only 10 different options. These include 504, 514, 524, 534, 544, 554, 564, 574, 584, and 594. I started dividing these by three and the highest possible option is 584 as 594 is not a solid number (no decimals). The possibilities for 37B are 370, 371, 372, 373, 374, 375, 376, 377, 378, 379. I have to divide by four and find the lowest number. The lowest number is 372 so if I subtract 372 by 584 I 212 which the largest positive difference

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  3. I think the largest positive difference between 5A4 and 37B is 212 , with 5A4 being 584, and 37B being 372. This was pretty much just guess and check, and I did it by dividing 584 by 4 (choosing a high number with 5?8), and found it was divisible by 4, and then did something similar with the other number (37B = 372). Then I subtracted the smaller number from the greater number to get 212.

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  4. To find the largest positive difference you have to make the larger number the largest it can possibly be and the smaller one, the smallest it can be.
    In this case you have to make 5A4 the largest it can possibly be and 37B the smallest.
    Since 5A4 is divisible by 4 the last 2 digits (A4 have to be divisible by 4) After experimenting you will find out that A will equal to 8 so 5A4 the largets it can possibly be is 584.
    Now for 37B it has to be divisible by 3 so we have to make B the smallest digit possible while making the sum of the digits add up to a multiple of 3. After experimenting I found out that 372 would be the smallest number possible.
    Now to find the difference 584-372=212 so the largest positive difference between 5A4 and 37B is 212.

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  5. Since the question is asking for the largest positive difference, we want 37B to be as low as it can be and 5A4 to be as high as it can be. Therefore, when we look at 5A4, the largest number we can find with 5 as the hundreds column and 4 in the ones column and is divisible by 4 is 584, as

    4x150= 600
    Working off that (subtract 4 every time)
    596, 592, 588, 584.

    584 is the highest number with the criteria given.

    For 37B, I based off 369:
    369, 372,375,378, 381.

    372 is the smallest number with the criteria and assumptions made.

    Difference: 584-372= 212.

    Therefore, the difference is 212.

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  6. To answer this question, I used divisibility rules that I know. First, I knew it had to be the largest positive difference, so that meant the highest possible number for the first one and and lowest for the second one. For the first one, a divisibility rule I know is that any number that ends in a two digit number that is divisible by 4 (12, 20, 24, etc.), that means the whole number is divisible by 4. Since it had to be the highest number possible and the second digit was the one that was unknown, I knew I had to find the highest 2 digit number that ends in 4 and is divisible by 4. I know that is 84 (84/4=21). That means that a=8. For the second number, it said that it was divisible by 3. A rule I know is that if the digits in a number add up to a number that is divisible by 3, it is divisible by 3. That made this number easy. I know that 7+3=10. The closest number to 10 (that is above it) that is divisible by 3 is 12, and 12-10 is 2. That meant that b=2. I had to find the difference between the numbers, so I just subtracted 372 from 584 and got the answer: 212

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  7. Here is how I solved it:
    So since 5A4 has to be divisible by 4 I did long division up until use are suppose to use A, now since after subtracting 4 from 5 I was left with one so A had to be a high number that when placed into the one's column beside 1 it would be divisible by four, and the only numbers that start with 1, are two digit and are divisible by 4 are 12 and 16 so since I want A to be very high so that I can increase the difference between 5A4 and 37B I chose to use 6 as A, so 564 is divisible by 4 so A=6.
    Now to find B once again I was left 1 after carry out the operations for long division and the only number that are divisible by 3 and are 2 digit and begin with 1 are; 12,15,and 18. Since I wanted B to be the lowest number (once again to increase the difference between the two numbers) I chose 2.
    So if 5A4 is now 564 and 37B is now 372, 564-372 is 192.
    The largest positive difference between 5A4 and 37B is 192.

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    Replies
    1. The reason why I got the wrong answer was because I only focused on two digit numbers above ten and below 20. Plus I used a method that wouldn't work in the end.

      Delete
  8. First, I look at what I know. The most critical piece of information provided is that I have to find the largest possible difference between two numbers, meaning that I have to make one of them as small as possible and the other as large as possible. I also need to keep in mind that the number 5A4 must be divisible by 4, and that the number 37B must be divisible by 3. We can see that 5A4 is larger than 37B (as in the hundreds column we have 5 and 3, and 5>3). To find A, I look at what the requirements for its value are. It has to be as large as possible, a single digit and must be divisible by 4. The largest single digit number divisible by 4 is 8, so A=8. Now finding B, I do the same thing, but know that B must make the number divisible by 3, and the lowest single digit that it can be. Therefore, B=2. So, my two numbers are 584 and 373. To find the largest positive difference, I subtract 372 from 584, getting 212.
    Therefore the largest positive difference between 5A4 and 37B (when 5A4 is divisible by 4 and 37B is divisible by 3) would be 212.

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  9. I used the method of guess and check, although I would love to hear how it could be done using algebra. Since, the question asks for the largest possible difference I subsituted A as the largest digits. Accordingly, I substituted the lowest digits for B. The only other requirement was that 5A4 had to be divisible by 4 and 37B had to be divisible by 3. Once again, using trial and error I found that A=8 and B=2. To double check, I went back and plugged in my identified values and saw they perfectly matched the requirements. 584 is divisble by 4 and 372 is divisible by 3. All in all, A=8 and B=2.

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    Replies
    1. In the end I subrtacted 372 from 584 to get the greatest possible difference of 212

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  10. I knew that to get the biggest difference the smaller number, 37B has to be as small as possible, while the larger number must be as big as possible.

    I know that to find a multiple of 4, the last two digits must be a multiple of four.
    If i need a multiple of 3 then all the digits must add up to a number that can be divided by 3.

    The biggest multiple of 4 under a hundred with a 4 on the one's digit is not 94, so the next is 84, which i a multiple of 4.
    That makes 5A4, 584

    The digits 3 and 7 add up to 10. to get to a number dividable by 3, you need to get to 12, by adding 2.
    So the digits are 372, meaning that 37B is 372.

    The difference between 584 and 372, achieved through subtraction is 212.
    So the greatest difference possible between he two number is 212.

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  11. I know to get the largest difference between the two numbers, I have to make the larger one larger, while making the smaller one smaller. I know that 5_4 is larger than 37_ as the number in the hundreds place is larger: 5>3 so I know in need to make 5_4 bigger.
    In addition, 5_4 needs to be divisible by 4.
    I started by using guess and check. Guess and check is a viable method as there is only 10 options: 0,1,2,3,4,5,6,7,8,9. So I started with 9 as it was the largest.
    594/4=148.5. It has a decimal so I try 8.
    584/4=146. So this is correct.
    A = 8
    Now B needs to be divisible by 3
    I try the smallest digit 0
    370/3=123.333333 repeating so it is incorrect.
    371/3=123.666667 so it is also incorrect.
    372/3=124 so it is correct.
    Therefore A=8 and B=2

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    Replies
    1. Oops, forgot to find the difference. 584-372=212
      Therefore the largest difference is 212.

      Delete
  12. To solve this problem, I used guess and check. I know that to find the largest difference, one number has to be bigger, and the other smaller. In addition, I can see that 37B is significantly smaller than 5A4, thus I started from the largest and smallest numbers between 1 and 9. Using trial and error, I found that A=8 and B=2. To double check, I plugged the new numbers back into the provided numbers, to make 584 and 372. I divided the two numbers by the required numbers. 584 was divisible by 4 and 372 was divisible by 3. Therefore, A is equal to 8 and B is equal to 2.

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  13. I used the method of guess and check as I believe that in this scenario it is the easiest. For 5A4 I know that in order for a number to be divisible by 4, the number formed by the last 2 digits have to be divisible by 4. So I tried 514 first but I know that 14 isn't divisible by 4 so then I tried 524 and since 24 is divisible by 4 I know that this is an option however it asks for the largest one possible so I continued. In the end I found that 584 is divisible by 4 and is also the largest one possible since 584/4=146. I then continued with 37B. For a number to be divisible by 3, the sum of all the digits has to be divisible by 3. So I did 3+7=10, so the options are 2, 5, 8. However since it wants this number to be the smallest i decided to use to so it is 372. Then I put the numbers back in and my numbers were correct. Therefore, B=2 and A=8.

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  14. Well, you need to find the largest difference, so one number had to be as large as possible, and the other had to be the smallest. Since 5A4 is already bigger than 37B, 5A4 would be larger, and 37B would be smaller.
    5A4:
    I replaced A with numbers 0-9, and tested to see which ones were divisible by 4 (into a whole number). 504, 524, 544, and basically every other number were divisible. 584 was the largest number i could get, so we have 1 number now.
    37B:
    I replaced B with numbers 0-9 (not really, as I stopped), and tested to see which ones were divisible by 3. 372 was divisible, and I stopped there. This time I wanted to get the lowest number, so there was no need to go on. Now I have both numbers.
    Here's my answer:
    Let A = 8
    Let B = 2
    The difference between them was 212.

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  15. I need to find the largest difference so I need to find the largest number that would make the number divisible 4 and the other divisible by 3.
    I can use numbers 0 to 9. 504, 524, 564 and 584 are all divisible by 4.
    Next I found letter B. I tested it with numbers 0 to 9 again and found that 2 was the lowest number. I had to get the largest number and smallest number to get the largest difference.
    So, My let statements are:
    Let A=8 and Let B = 2
    And 584-372=212
    Therefor, Let the difference = 212

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  16. The largest difference is 212. How I did this was that I first looked at the digits I can use to replace A in 5A4. The digits that I can use range from 0 - 9. So 504, 514, 524, 534, 544, 554, 564, 574, 584, and 594. So first I looked at the biggest number which was 594 to see if it was divisible by 4. It wasn't so I moved on to the second largest number, 584. Then I found that the answer was 146 so that meant that 584 was divisible by 4. Then I moved on to 37B. So once again I knew that the digits that B could be range from 0-9. So the number could be 370, 371, 372, 373, 374, 375, 376, 377, 378, and 379. So I started with 370 at first and that wasn't divisible by 3, so I moved on to the next number. 371 wasn't divisible by 3 either so I moved onto 372. In the end 372 was divisible by 3. Then I did 584 - 372 to find the difference as stated in the question. So 584 - 372 is equal to 212. Thus, 212 is the largest difference between 5A4 and 37B.

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  17. Oops, I didn't post a comment in time. But the largest difference between 5A4 and 37B is 212.
    Let A equal 8 and let B equal 2.
    To solve this answer, I just did long division as if the variables were not there. Until, I had to bring down, the variable, I would just do regular math. The question stated that 5A4 was divisible by 4, so I did just that, 5A4/4. If you were to do the division, you would have 1 and you would have to bring down the A. So what number in the 0-9 range should be used to be brought down to make a number in the 10-19 range? Well to help me, I used the number after A, 4 so that meant that when 1A is divided by 4, 4 can be divisible by it (because the that would just mean I have to do 4/4 next), the quotient of 1A/4 subtracted from 1A leaves me with a difference of 2,4, or 8 (because 24, 44, and 84 are all divisible by 4). I also have to note that I need to find the greatest difference between 5A4 and 37B. That would mean I would have to choose the largest number possible for 5A4 to get the greatest difference. If I chose A to be a number that when combined with 1 (making it 1A) equals to be a number that is divisible by 4, it would be 6 (because 16 is divisible by 4 and is the largest number divisible by 4 in the range of 10-19). If I were to choose the other option (1A-(1A/4)), the largest option would be for A to be 8 because 584/4 is equal to 146). So now I have A.
    To find B, I did a similar method, but I divided 37B by 3 because the question states 37B is divisible by 3. When doing the division, I have 1 left before having to drop the B. If I'm trying to find the largest difference, 37B must be as small as possible. B is also the last number of 37B which means that 1B must be divisible by 3. My smallest option, B is 2 because 12 is the smallest number divisible by 3 in the range of 10-19. So that means if B is 2, 37B is 372 and 372/3 is equal to 124.
    Now to actually find the answer. To find it, I subtract 372 from 584 to get 212, the largest difference.
    Sorry if this didn't make sense, my method was hard to explain in words.

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  18. First I found what A and B were. to get the largest difference, needed the smallest possibility for 37B and the largest for 5A4. I know the unknown can only be one digit and has to be divisible by a certain # (3 or 4). I concluded that A = 8 and B = 7, so since the difference between 372 and 584 is 212, that is the largest positive difference.

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