subhashghosh wrote:

HI Bunuel

What does this mean ?

remainder upon division the power 4+4x by cyclicity 4 is 0, which means that 3^{(4+4x)} will have the same last digit as 3^4)

Regards,

Subhash

Theory:

First of all note that the last digit of xyz^(positive integer) is the same as that of z^(positive integer);

• Integers ending with 0, 1, 5 or 6, in the positive integer power, have the same last digit as the base (cyclicity of 1):

xyz0^(positive integer) ends with 0;

xyz1^(positive integer) ends with 1;

xyz5^(positive integer) ends with 5;

xyz6^(positive integer) ends with 6;

• Integers ending with 2, 3, 7 and 8 have a cyclicity of 4;

For example last digit of

xyz3^{positive \ integer} repeats in blocks of 4: {3, 9, 7, 1} - {3, 9, 7, 1} - ... So cyclicity of the last digit of 3 in power is 4.

• Integers ending with 4 and 9 have a cyclicity of 2.

Now, last digit of

xyz3^{positive \ integer} repeats in blocks of 4: {3, 9, 7, 1} - {3, 9, 7, 1} - ... means that:

3^1, 3^5, 3^9, 3^13, ..., 3^(4x+1), all will have the same last digit as 3^1 so 1;

3^2, 3^6, 3^10, 3^14, ..., 3^(4x+2), all will have the same last digit as 3^2 so 9;

3^3, 3^7, 3^11, 3^15, ..., 3^(4x+3), all will have the same last digit as 3^3 so 7;

3^4, 3^8, 3^12, 3^16, ..., 3^(4x), all will have the same last digit as 3^4 so 1;

So to get the last digit of 3^x, (where x is a positive integer) you should divide x by 4 (cylcility) and look at the remainder:

If remainder is 1 then the last digit will be the same as for 3^1 (so the first digit from the pattern {3, 9, 7, 1});

If remainder is 2 then the last digit will be the same as for 3^2 (so the second digit from the pattern {3, 9, 7, 1});

If remainder is 3 then the last digit will be the same as for 3^3 (so the third digit from the pattern {3, 9, 7, 1});

If remainder is 0 then the last digit will be the same as for 3^4 (so the fourth digit from the pattern {3, 9, 7, 1});

Next, as 4+4x (the power of 3^(4+4x)) is clearly divisible by 4 (remainder 0) then the last digit of 3^(4+4x) is the same as the last digit of 3^4 so 1.

You can apply this to integers ending with other digits as well (with necessary modification of pattern).

Hope it's clear.

P.S. Check this for more:

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