Multi String Pearl Necklace

Mathematical Games "magic" and group structure


symmetries presented by the group algebraic structure can support an infinite number of mathematical games, magic .

Imagine the operation we will call Alpha Expression [7.1000000]: I raise the number 7 to 3x10 ^1000000 exponent, then most all the figures of the result, that amount and divide it by 9 am the rest (Note **) (module 9). If I say to that number, millions of numbers, the result is 1 and also I can assure you without even attempting the operation may seem almost a magic trick but in reality, the result is almost as simple as raising the unit to any finite power and as big as we want: the result is always 1.

The key to this apparent mathematical magic trick the algebraic structure is very important, group call, about which we have spoken sometimes (Ver1) (View 2) .

group module 9 of the sum of digits of a set of numbers with the product.
establish an internal operation (product) between the elements of a set {1,2,4,5,7,8} and observe the corresponding graph of group structure, which shows the relationship of each element with all others (1,2,4,5,7 and 8 values \u200b\u200bare not arbitrary, corresponding to the remainder (mod 9) of the sum of the numbers of each of the prime numbers that exist (except ***), Note the 3):



In view of the results showed some interesting properties:

Sum.mod.9 (2 ² ) = 4.
Sum.mod.9 (2 3) = 8.
Sum.mod.9 (2 4) = 7.
Sum.mod.9 (2 5) = 5.
Sum.mod.9 (2 6) = 1.
Sum.mod.9 (2 1) = 2.

This means that the 2 with the product operation is capable of generating all elements of the group. You can see that at 5 the same thing happens. Moreover, we find that:

Sum.mod.9 (2 6) = 1. Expression (a).
Sum.mod.9 (4 ³ ) = 1. Expression (b).
Sum.mod.9 (5 ⁶ ) = 1. Expression (c).
Sum.mod.9 (7 3) = 1. Expression (D).
Sum.mod.9 (8 2) = 1. Expression (e). Since

Sum.mod.9 (1 n) = 1, n whatever finite find that each of the expressions (a), (b), (c), (d) and (e ) could give an account finite, but arbitrarily large expressions like Alfa expression we used at the beginning of this post . For this, in particular, have taken the expression (d) that we have raised to the power 1000 000.

Note (**) 1000 000 Instead of going to see a couple of simple examples with exponents more affordable. For example exponent 3x10: 7 ³⁰ = 22539340290692258087863249. The sum of the numbers divided by nine (mod 9) gives the remainder "1".,

Another example, an example 3x7: 7 ²¹ = 558545864083284007, etc. Or the simplest of all 7 ³ = 343. The sum of the numbers is 10, and 10 / 9 is a division of rest "1."
easier to see: any number consisting of the unit followed by zero raised to any finite power unit will always followed by zeros, so the sum of its digits will always be unity. It is clear that in this case is no different, with numbers whose sum of digits and also "1" (module 9): 91 and 19, 28 and 82, 46 and 64 ... For example: 9 = 6,240,321,451, the sum of the numbers gives 28, 28 / 9 = 3, with other "1." With the other numbers in this list occurs equally.

Note (***): The distribution of primes less than 1000 compared to the rest module 9 of the sum of its digits (except 3), out of curiosity, is this: Rest 1 (27), other 2 (30), other 4 (26), 5 rest (29), other 7 (26) and other 8 (29). In total 167 primes.