rros0770 wrote:Hey Six, you lost me with that one.
I'm going to assume that the (%) sign I'm seeing is meant to represent division?
(7^12x * 7^3) / 5 looks like it was factored correctly, but I'm not quite sure how you converted that into (2^12x * 2^3) / 7. I'm guessing you obtained this by logic, as it doesn't seem to follow any exponent rule I'm familiar with (correct me if I'm wrong)? Just curious how this was derived because this quickly arrived at the correct answer of remainder = 1.
I came to remainder 1 using the same method as Arcane. Except I just focused on multiplying the units digit by 7 each time because that's all we'll need to know to determine the repeating pattern...
Hi!
What
sixpointer wrote in his reply was an application of Modulo Arithmetic. The % (percentage) sign is actually the modulus sign.
(a mod n) = a%n = b, means when a is divided by n, the remainder is b.
Some basic theorems of modulo arithmetic are:
1. (a + b) mod n = [(a mod n) + (b mod n)] mod n
2. (ab) mod n = [(a mod n)*(b mod n)] mod n
3. a^b mod n = [(a mod n)^b] mod n
These formulas can be proved very easily. In fact, (1) is very much trivial.
Now using these formulas here,
[7^(12x + 3) + 3] mod 5
= [7^(12x + 3) mod 5 + 3 mod 5] mod 5
= [[(7 mod 5)^(12x + 3)] mod 5 + 3 ] mod 5
= [[2^(12x + 3)] mod 5 + 3] mod 5
= [[8*(2^12x)] mod 5 + 3 ] mod 5
= [[(8 mod 5)*(2^12x)] mod 5 + 3 ] mod 5
= [[3*(2^12x) mod 5] mod 5 + 3] mod 5
= [[3*(4096^x) mod 5] mod 5 + 3] mod 5
= [[3*(4096 mod 5)^x mod 5] mod 5 + 3] mod 5
= [[3*1] mod 5] mod 5 + 3 ] mod 5
= [3+3] mod 5
= 1
Though the alternate solution provided to this question is standard and easy, this method will help you to solve this kind of problems real fast. (Provided you have mastered it!
)
Hope this helps.
