Blog Post No.88: The magic of 4096

4096 is well known in the computer world as 2 ^ 12 (64 x 64) and that's pretty straight-forward

Anagram of 4096 is 9604 which is also a perfect square - 98 x 98

The above are fairly straight-forward and simple but check out this one

4096 x 4096 = 16777216

Now carefully dissect the 8 digit number - 16 and 16 are perfect squares of 4 which is the first digit of 4096

7772 is 4 x 67 x 29 - Look at this again - 4 is the first digit of 4096 and the sum of other two factors (67 + 29) is 96 which are the last two digits of 4096

Not many squares exhibit this kind of a pattern and if there are few more would love to see that listed down

Blog Post #87: Attributes of 1312

During one of our conversations one of my pretty close school mate, Ram, said that his favourite number is 1312 and wanted to check if there are any cool attributes associated with this number

Here are few to start with

a) 1312 is sum of two perfect squares - 1296 + 16

b) 13 + 12 and 13-12 are perfect squares again

c) 1312 can be written as 1000 + 156 + 156 which is 10 ^ 3 + 13 x 12 + 13 x 12 - not many numbers exhibit this property where it is a sum of perfect cube (1000) added with a number (156) whose factors (13, 12 here) are sub-digits of the result! (1312)

d) Now this gets interesting - Look at the odd anagrams of 1312

1123 - Prime
1213 - Prime
1321 - Prime
2113 - Prime
2131 - Prime
2311 - Prime
3121 - Prime
3211 - Only exception but the factors are 13 and 19 which are prime numbers again and 13 is the first two digits of 1312!

e) 2131 is the mirror image of 1312 - 21 x 31 = 651 which is the mirror image of 156 (13 x 12)

Ram - over to you to jot down few more :)

Blog Post #86: Cool attributes of the number 12544

To start with 12544 is a perfect square and I always find perfect squares having lot of unique attributes

a) 12544 - 112 x 112
b) 125 - 44 = 81 which is 9 x 9
c) 125 + 44 = 169 which 13 x 13
d) Now 9 + 13 = 22 and 13 - 9 = 4; Now look at the square root which is 112 (11 + 11 = 22 and 2 + 2 = 4)

How many squares exhibit this kind of pattern? Can we explore further?

Few more interesting attributes

e) Go back to 12544 and add some of the digits: 144 + 25 = 169 which is again 13 x 13 and 144 is 12 ^ 2 and 25 is 5 ^ 2 (12 and 5 are the first 3 digits of 12544)
f) Now look at 112 again - 11 x 11 = 121, 2 x 2 = 4. 121 + 4 = 125 (which is the first 3 digits of 12544 again)