Number patterns

Number theory has amazing patterns. Well, here I’ll give some interesting ones on sums of digits.

Sum of digits (s.o.d) of a natural number is found simply by computing the number modulo 9. (eg. s.o.d 157= 4)

Consider powers of  2. The sums of digits of the powers follow an interesting cycle- 1, 2, 4, 8, 7, 5.

1*2 = 2

2*2=4

4*2=8

8*2=16=7 (mod 9)

16*2=32=5 (mod 9)

32*2=64=1 (mod 9)

And again the same cycle repeats after 64. (s.o.d 128 = 2; s.o.d 256 = 4, and so on.)

Well, intuitively the proof is simple.

It follows from the result that s.o.d ( product of 2 numbers) = s.o.d of 1st number * s.o.d of 2nd number.

i.e. if we have 2 natural numbers, x and y; then s.o.d (x*y) = s.o.d (x) * s.o.d (y).

The result on product of s.o.d of numbers can be broken down further; that is, the above result can be obtained from the fact that s.o.d ( sum of 2 numbers) = s.o.d of 1st number + s.o.d of 2nd number.

From the result on product of sums of digits, the pattern stated above for powers of 2 is easily obtained by induction.

Well, we can state similar pattern for different numbers:

The cycle for the powers of 5 is 1, 5, 7, 8, 4, 2.

Now, we can see an interesting design- the cycle for powers of 5 is “opposite in direction” to the cycle for powers of 2.

Well, this is actually to be expected, and one can easily see intuitively why this is so. I leave it for the reader to figure it out. (Hint: 2^n * 5^n = 10^n = 1 (mod 9) ).

So the cycles for 2 and 5 are of lengths 6 each. The cycle length for powers of 7 is 3 (Cycle is  {1, 7, 4}.)

The cycle for 11 is 1, 2, 4, 8, 7, 5. Hey, this is the same as that for 2. Why? 11= 2 (mod 9). That’s why. If 2 numbers are equal modulo 9, they will have the same cycle, right.

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