I am a quadriplegic. When I wake up in the middle of the night, I can’t just grab a book, flip on the TV, get on my computer, go for a walk, or most any other alternative to lying awake that anyone else might choose. As a result, I came up with one way of amusing myself and I thought I would share it with you.

As you will likely guess from the title, my form of amusement involves doing mathematics with the number that appears on my digital clock that is visible from my bed even at night. Basically, it involves looking at the clock number and trying to figure out as quickly as I can what numbers will divide into it without fractions.

A few ground rules will make this explanation easier to write and easier to follow. When I say clock number it is the number which appears on the clock face. For the purposes of this piece 4:08 will be expressed as 408. We will be dealing with the numbers as they appear rather than converting them to a number of minutes; thus 6 o’clock will be 600 and not 360 minutes. Finally, if a number is divisible by any odd number then if the number is even it is also divisible by twice that odd number.

Now that we have all that out of the way, I will give you some tricks and tips for determining fairly quickly whether a number is divisible and by what numbers. Some of these will work for any number and some are limited to clock math.

Of course, any number is divisible by 1, and any even number is divisible by 2.

3, 6. Any number for which the digits add up to a multiple of 3 is divisible by 3. If the number is even it is also divisible by 6.

4. Any number of any size is evenly divisible by 4 down to the hundreds digit. This is because 100 is evenly divisible by 4. Therefore, all you have to do is look at the last two digits (for a clock number, this is the minutes). If the last two digits are divisible by 4, the whole number is divisible by 4.

5, 10. Any number ending in 5 or 0 is divisible by 5. If the number is an even number it is also divisible by 10.

7, 14. There is no math trick of which I am aware to quickly show whether any number is divisible by 7. However, with clock math there is a trick you can use. 7*14 is equal to 98. This is 2 short of 100. Therefore, for any hour you can multiply 2 by the hour and that is how short you are of reaching that hour; you then add the multiple of 7 necessary to get you over the hour. Example: for 6 o’clock, 6*2 is equal to 12, therefore you can add 14 and your starting point is 602. You may then subtract 602 from any time during the 6 o’clock hour and see if it is a multiple of 7, and of 14. For example: subtracting 602 from 630 leaves 28. Therefore, 630 is divisible by 7 and 14.

8. 8 is similar to 4 but with a twist. Any even hundred (2, 4, 6, 8, and 0) is evenly divisible by 8; therefore all you need to do is look at the last two digits and if they are divisible by 8 then the whole number is divisible by 8. However, this does not work for odd hundreds (1, 3, 5, 7, and 9); for each of these numbers you need an extra 4 for them to be divisible by 8. That means for odd hundreds the last two digits must be divisible by four but NOT divisible by 8. Examples: 744 and 648 are divisible by 8, 644 and 748 are not divisible by 8.

9, 18. If the digits of any whole number add up to a multiple of 9 then that number is divisible by 9; if the number is an even number it is divisible by 18.

11, 22. There is a very interesting trick for dividing by 11 which actually works for any number of any size. I call it “folding”. It involves taking the ones digit off the number and adding it to the hundreds digit. With clock numbers you normally only have to do this one time but with larger numbers it can literally be done all the way up the line. To illustrate let’s take the number 1243. You remove the 3 and added to the 2; this gives you 154, which if you know your “times tables” you know is 11*14. Believe it or not, this actually tells you that 1243 is divisible by 11. If the number is even, such as 1254, is also divisible by 22. Construct any number of any size, taking care that you know whether it is truly divisible by 11 and then try this technique; it will work.

12. To divide by 12 we need to borrow a technique from the world of fractions. Steady now! Take a few deep breaths and keep reading. The technique we are borrowing is that of the Least Common Denominator or LCD for short. Specifically, if any number is divisible by 3 and 4 then it is also divisible by 12, because 12 is the LCD of 3 and 4. This technique works for any number where you have two or more different numbers by which the number may be divided. Take note however that they must be different numbers; 3 and 9 have an LCD of 9, and 4 and 8 have an LCD of 8. While not the same numbers, they are multiples of each other and don’t get you anywhere.

13. There is no easy way to get divisible by 13 for any number. For clock numbers it works to know that 7*13 is 91 and 8*13 is 104. You can use either number times the number of the hour to get a starting point. I usually take 9 times the hour and then add the needed multiple of 13 to get over the hour (example: 6*9 is 54, which requires adding 65, giving you 611 as your starting point for the 6 o’clock hour). After that, of course, you simply subtract your starting number from the clock number to see if you have a multiple of 13.

15, 30. If any number is divisible by 3 and 5 then it is also divisible by 15; if it is an even number it is divisible by 30.

16. There is not an easy way to determine divisible by 16 for any number. However for clock numbers you can get by remembering that 6*16 is 96. That makes your start numbers for the hours 112, 208, 304, and 400. The pattern starts again with 512 through 800, and 912 through 1200.

17, 34. For clock numbers you can remember that 6*17 is 102. Thus, 102 times the hour gives you your starting point for each hour (starting with 918 you can go back 17 as well is forward).

19, 38. For clock numbers you can remember that 5*19 is 95. Therefore take 5 times the hour to determine what multiple of 19 you need to add to get your starting number for the hour (example 6*5 is 30, so you add 38, making 608 your starting point).

20. Here we are back to using the LCD; if a number is divisible by 5 and 4 then it is divisible by 20.

21, 42. We use the LCD for this number as well; if the number is divisible by 7 and 3 then it is divisible by 21.

23, 46. For clock numbers, be aware that 4*23 is 92; you may take 8 times the hour and add the multiple of 23 needed to cover it (example: 6*8 is 48, meaning you need to add 69 to cover it, making your starting number 621).

24. Using the LCD for this number tells us that if any number is divisible by 8 and 3 that is also divisible by 24.

25, 50. Numbers which are divisible by 25 and 50 are relatively easy to spot, so I’ll leave it at that.

Beyond this point, I usually don’t try to do clock math with numbers 27 and above. The math, except where you can do an LCD (such as 28 from 7 and 4), gets too time-consuming to be worth it.

I hope you found this dissertation, if not entertaining, then at least enlightening; perhaps telling you some math tricks you did not know. Thank you for taking the time to read.

The End

Michael,

Fascinating…a great exercise for any insomniac.

I love it!

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Thank you Sarah. It usually works a lot faster than counting sheep.

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It sure does!

I was in the process of adding a picture yesterday and was interrupted

by a phone call. Sorry ’bout that…

btw, looks like 3 and 9 on the one I chose.

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Thank you Sarah, and that’s no problem. I appreciate the addition. I did a quick run through and I believe you are correct; 11:43 is divisible by 3 and 9. Other possibilities fell on either side of that number.

Looking through the text, I noticed a couple of “jokers” my Dragon software slipped in on me. They are minor, so I think I will leave them alone. However I noticed that where my original had a special “X” symbol for multiplication, when I pasted it here it turned into a “/”. I guess the symbol that Microsoft Word used is not one that translates in WordPress.

Do you have an easy way to change all of those instances? If not, I will have to go through and update them myself, because that could easily be confusing.

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Michael,

I can change them. The accepted symbol for digital multiplication is the asterisk.

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Oops, I see you have changed them.

Often times the ‘x’ is taken for ‘X’ rated. : )

However, the changes you have made are perfectly clear in their meaning.

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Actually, Sarah, I had nothing to do with that. It looks like the same symbol Microsoft Word was using suddenly reappeared in my text. However, I had not been back to this page since leaving that reply until just now. I saw both of your comments at once, and had to look to see that the symbol had indeed changed.

(Hears spooky music) Do you believe in the WordPress Fairy?

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I think instead of spooky music, I must’ve been hearing the Twilight Zone theme. Whatever magic the WordPress Fairy did apparently wore off; the “/” is back as I am looking at it now.

If you can do a “replace all” and replace all of the symbols at once, please do. It would take me a little doing to update the entire text. On the other hand, if you would need to do the replacements one at a time yourself, let me know and I will take care of it rather than put the work off on you.

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Sarah, I went ahead and took care of this myself because I spotted something else that simply had to be changed. In the early part of the text I said that if any clock number was divisible by an odd number then if the clock number was even it was divisible by twice that number. That actually works for any number. I made the same mistake when saying that if the digits of the clock number were divisible by 3 then if the clock number was even it was divisible by 6. Again that is true of any number. I think you can see why I felt the need to get that changed.

Thanks for all your help, and I mean that.

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