Thursday 29 December 2011

Repeated Divisibility

Let me tell about a 17 digit number,which I found after several attempts,having a special property.
The number is 34880 44860 36585 69.
This number is divisible by 17.
When you delete the last digit 9,the new 16 digit number will be divisible by 16.
Continuing,when you delete the last digit 6 from this 16 digit number,the new 15 digit number will be divisible by 15 and so on,showing divisibility by 14,13,12,11....etc
I am showing below some of the quotients relating to the  divisions-
Division by 17-D/17...Quotient is 20517 91094 332857
D/16...   Number to be divided is 34880 44860 365856 .Quotient is  21800 28037 72866
D/15...   Number to be divided is 34880 44860 36585.Quotient is  23253 63240 2439
D/14...   Number to be divided is 34880 44860 3658...Quotient is 24914 60614 57
D/11..... Number to be divided is 34880 448603.Quotient is 31709 49873
D/7....... Number to be divided is 3488044.....Quotient is 498292
D/4....... Number to be divided is 3488 ..Quotient is 872
It is difficult to work out such numbers.While you can find many numbers upto 9 or 10 digits having divisibility by numbers from 2 to 10,it may be difficult to get the 11th,12th etc digits and then you have to start all over again..  

Saturday 24 December 2011

Curious facts about square numbers

Let me present some curious facts about square numbers-
1)Two square numbers when placed side by side will reveal another square number-Examples
Square numbers 1 and 225 will produce 1225 which is the square of 35.
Similarly 4 and  225 will produce 4225-square of 65
16 and 81 will produce 1681-square of 41
144(sq.of 12) and 2401(sq of 49) will produce 1442401(sq of 1201)
2)In the following cases the square numbers follow a pattern-
7*7=49...67*67=4489....667*667=444889.....6667*6667=44448889
4*4=16.. 34*34=1156....334*334=111556.....3334*3334=11115556
3)The sum of 20 and 25 is 45 while the square of 45 is 2025.
Similarly the sum of  30 and 25 is 55 while the square of 55 is 3025.
The sum of 494 and 209 is 703 while the square of 703 is 494209
The sum of 88 and 209 is 297 while the square of 297 is 88209.There are many more such cases
4)The square of 12 viz 144 and the square of 33 viz 1089 will add up to1233
5)The square of 13 is 169 ,while the square of 31 is 961
Similarly the square of 12 is 144,while the square of 21 is 441.
The square of 102 is 10404,while the square of 201 is 40401.
6) The square of 13 is 169 while 1,6 and 9 will add upto 16,As a surprise the square of 16 is 256 and the digits 2,5 and 6 will add up to 13.
7)The square of 45 is 2025.If you add 1 to each digit in 2025 you will have 3136 which is a square of 56
Similarly the square of 34 is 1156.If you add 3 to each digit in 1156 you will have 4489 which is a square of 67.
8)Though the dates are too far away even some dates will reveal square numbers if you remove the hyphens as shown below-
4-01-6016 ...the square of 2004 being 4016016
Similar cases date 1-01-6064....sq of 1008 being 1016064
Date  1-06-5024...square of 1032 being 1065024.There are also many more cases.

Tuesday 20 December 2011

Divisibility tests for 7,11 and 13

You may be aware of divisibility tests for numbers like 3 ,4,5,8,9,16 etc.
I now present you with a combined divisibility test for 7,11 and 13.
I am taking the number128704719 for the test.
You divide it into groups of 3 digits starting from the right.Call the groups A,B,and C.Thus A=719,B=704 and C=128.Add up the figures in alternative groups and find the difference between those sums.Thus A+C=847.For B there is no further group to add and so the sum is the same number as B viz 704.The difference between the sums is thus 847-704=143.Test this result for divisibility by 7 ,11, and 13.If it is divisible the original number will also be divisible.As 143 is divisible by 11 and 13 and not by 7 the original number 128704719 will be divisible by 11 and 13 and not by 7.
Actually 128704719= 11 times 11700429 and 13 times 9900363 while division by 7 gives a quotient 18386388 and a remainder 3.
When you take a number with say10 or 11 digits for test you will have a group D which can be added to B
This method is based on the fact that the product of 7,11 and 13 is 1001.
If the factors of numbers like 1001 are known,you can find similar procedures for testing .Example 1003 has the factor 17.-can be used for testing divisibility by 17....1007 has the factor 19-can be used for 19.. . 

Friday 16 December 2011

Calculation Tricks

In my blog relating to Fun with numbers,I mentioned about cyclic number 142857.When you multiply this number with 2,3,4,5 and 6,you will get 285714,428571,571428,714285 and 857142 with the digits readjusting their places but following the same pattern left to right.When you multiply the number with 7 you will get 999999.
It is easy to remember /realise the correct answer by remembering what should be last digit in each case.i.e when you multiply with 6,you must have 2 as the last digit because 6 times 7(the last digit in 142857) is 42.etcetc.
Now let me tell you a calculation trick using these results.One will be able tell immediately or in the beginning a few minutes later what the result will be when you multiply the number say with 16 or,345.or 6305 or any other number.
In fact the results are 2285712,.....49285665     900713385....Let me explain how these results become easy to find.
As you get 999999 by multiplying with 7 viz 'one million less one' you should get 2 million less 2 when multiplying with 14,....8 million less 8 when multiplying with 56 (which is 8 times 7) etc etc.
Now as 16 equals 14 plus 2 (viz 2 times 7 plus 2),you can find the results of multiplication with 14  and 2 and add them together for getting what the result should be when multiplying with 16.viz it should be 2 million less 2 plus 285714 i.e 2285714 less2 i.e 2285712 as mentioned earlier by me.
Similarly for 345 you will notice by dividing it with7 that it is 49 times 7 plus 2,So the result of multiplication with 345 is obtained by adding the results of multiplication with 49 times 7  and the result of multiplication with2  viz it is 49 million less 49 plus 285714 viz 49285714 less 49 viz it is  49285665 as mentioned by me.
For 6305 you can see that it is 900 times 7 plus 5 so the result should be 900 million less 900 plus 714285 viz it is 900714285 less 900 i.e 900713385 as mentioned by me.
The calculations would seem to be difficult for a beginner but by practicing with smaller numbers first one can easily master the technique.
The result of multiplication with say 23641286 will be  3377326 million less 3377326 plus 571428 viz 3377323194102.
Read the complete process a couple of times to get yourself   familiarised  with the technique.                                                                                                                                                      ..... 

Tuesday 13 December 2011

Black hole number 123

This is about the number 123 ,which acts like a black hole.
Take any natural number which can be regarded as a string such as  9288759..Count the number of even digits,the number of odd digits and the total number of digits.These are 3(evens),4(odds) and 7(total number of digits).Use these three to form the next number or string viz 347.Repeat the process with 347 to get 1,2 and 3 to get the next number/string as 123.Repeat the process again ,you will get the same number 123.There is thus no escape from this number.
You can start with any other number  and in every case the process when continued as many times as are required will end up with 123.
Example-65498299877543---Evens(6),Odds(8),Total (14)
Next number is thus 6814.....Evens(3),Odds(1),Total(4)
Next number 314.....Evens(1),Odds(2),Total (3)...leading to the black hole number 123.
  

Sunday 11 December 2011

Digits being same

Let me introduce 2 sets of numbers
First set 13458
Second set 02679
Let me show you how the numbers can be made to travel.With each set I make 18 five digit numbers .You can see that the numbers in the second set are exactly double those in the first set.
13485/26970.....13548/27096....13845/27690....14538/29076.....14835/29670.....14853/29706
31485/62970.....34851/69702....35148/70296....35481/70962.....38145/76290..... 38451/76902
45138/90276.....45381/90762.....48135/96270...48351/96702.....48513/97026...   48531/97062   ....