Is there a pattern/ rule for 7 times tables? Why does the rule/ pattern work?

Discussion in 'Mathematics' started by Libstar, Jan 18, 2011.

1. Hello,

This is something I occasionally think about as other numbers have patters:

1 - every integer

2 - even numbers

3 - add digits together until single and get 3,6, or 9

4 - for double digits if tens digits is even then ones digits is 0, 4, or 8, if odd then 2 or 6

5 - ones digits is 0 or 5

6 - is an even number and add digits together until single and get 3,6, or 9

7 - ?

8 - sum of digits adds to one less than previous (anything better?)

9 - sum of digits until single - get 9

10 - ends in zero

There is a pattern in sevenths but in sevens?

2. There is nothing as handy as the divisibility rules for multiples of 2, 9, etc.
[And a quick glance at Wikipedia will show this - probably not worth bothering with because for numbers below a million it is quicker just to divide by 7.]
Multiples of numbers-that-end-in-7 are as far below a multiple of 10 as numbers-that-end-in-3 are above a multiple of 10.
Compare the 7x table and the 3x table, but do the 3x "in reverse" (starting at 9x3 and finishing at 1x3):
7x 3x
1x7 = 7 9x3 = 27
2x7 = 14 8x3 = 24
3x7 = 21 7x3 = 21
4x7 = 28 6x3 = 18
...
Look at the units digits for both.

3. From memory of a long time ago :
Take the last digit off you number
Double it
Subtract it from whats left
Does the number now divide by 7?
Example
819
remove and double last digit, therefore double 9 = 18
subtract the 18 from 81 (the remaining number)
= 63
That divides by 7, therefore 819 is divisible by 7
Another one
14
take the 4 off and double it
leaves you 8
subtract that from the remaining 1 and you have neg 7

take the

4. Betamale is correct.
As well as the "x2 and subtract" rule there is the complementary "x5 and add" rule which works in a similar way.
Testing 108 for divisibility by 7: Split this to make 10 and 8
Last digit x5 gives 40
40 + the truncated number (10) gives 50, which is not a multiple of 7.
Therefore 108 is not a multiple of 7.
An explanation of this and other rules can be found here.
Andrew Jeffrey showed me another great trick with the 7 times table.
Draw a conventional "Noughts and Crosses" Grid (3 by 3)
In the first row, write the numbers 1, 2, 3
In the second row, write the numbers 2, 3, 4
In the third row, write the numbers 4, 5, 6
These represent the TENS digits.
Now go to the RIGHT hand column and write DOWNWARDS to insert the UNITS digits.
In the right column, write the numbers 1, 2, 3
In the middle column, write the numbers 4, 5, 6
In the left column, write the numbers 7, 8, 9
If all of that has gone correctly you will be looking at the 7x table.

5. The easiest rule for smallish numbers is try short division. For large numbers it is helpful to note that 1001 is divisible by 7 (11 and 13 also), and this observation can be worked up into a rule that slightly reduces the effort. I still prefer short division.

6. Thankyou to everyone who gave up their valuable time to answer my Q. I really appreciate it.

7. Don't know if this will help looking at divisibility by 7.
Take the last digit of the number and double it. Subtract it from the rest of the number. If this answer is divisible by 7 (including 0), the original number is divisible by 7. If the answer is too large you can apply the rule again until the answer is small enough for you to know whether it is divisible by 7.
Example 1. 203 Double last digit 3 + 3 = 6. Take answer from rest of the number 20 - 6 = 14.
14 is divisible by 7, therefore 203 is divisible by 7.
Example 2. 2478 Double last digit 8 + 8 = 16. Take answer from rest of the number 247 - 16 = 231.
231 Double last digit 1 + 1 = 2. Take answer from rest of the number 23 - 2 = 21.
21 is divisible by 7, therefore 2478 is divisible by 7.

8. Nice one.
Never seen it before, but I do know that (1 - (-20)) is divisible by 7.
Keep them coming.

9. That isn't really one for them to rely on remembering, but good practise for putting numbers through an algorythm.

10. This works with a telephone key pad layout and helps with x tables of numbers in the corners (similar to maths126 suggestion possibly) so is helpful for 7 x tables. Start in the corners going clockwise ( e.g. 123) then go in same direction for the next row ( 456) and so on, going to the 0 last. When numbers go over 9 you add a ten. 1 x table is easy though.
1 2 3
4 5 6
7 8 9
0
When you do it for 3 7 and 9 the patterns continues. You can add in the tens at the side and the pattern for the tens for the 7s is
2 4 6 then 9 11 13 etc
1 3 5 8 10 12
0 2 4 7 9 11
7 14
If you overlay this in front of the top grid you've got your 7 x tables and can work out large multiples at a glance or very easily without multiplying anything.

11. Forgot to say that the same thing works for the 2 4 6 and 8 x table if you set it out:
2 4
6 8
0
and working from the corners in a clockwise direction initially and then continuing in that direction for the next row,
e.g. 2 4 6 8 0 (obvious)
4 8 2 6 0 (4 8 12 16 20
8 6 4 2 0 (8 16 24 32 40)
6 2 8 4 0 (6 12 18 24 30)

12. There 'is' a pattern to the 7 times table if you do the 'casting out' of the answers ( you can do this for the other times tables to see other patterns)

1 x 7 = 7 7
2 x 7 = 14 1 + 4 = 5
3 x 7 = 21 2 + 1 = 3
4 x 7 = 28 2 + 9 = 10 1 + 0 = 1
5 x 7 = 35 3 + 5 = 8
6 x 7 = 42 4 + 2 = 6 etc etc
I will leave you to do the rest yourself.
You probably know the Cyclic patterns for : 1/7,2/7. 3/7 ...............etc ?
Best regards
Phil Chan
Longdean school -Hemel Hempstead

13. shouldn't that be 0, 1, 2 in the first row?

14. and another thought - i can see mathswife's point that betamale or maths126's divisibilty tests are good examples of repeated algorithms when used for large numbers,but if you really wanted a divisibilty test for a 4 or 5 digit number, in practise it would be easier to work modulus 7, sort of
so 819 &equiv; 112 &equiv; 42 is divisible by 7
108 &equiv; 38 is not divisible by 7
however - i was asked today if i knew any patterns to the 7 times table - and wasn't i able to show off cheers guys