Can anyone help with this math question?

Question

Find the remainder when nineteen to the power ninety-two is divided by ninety-two?

Answer 1

Just use modulo n. By the Euler-phi theorem,
x^phi of a = 1 mod a where x and a are relatively prime.
modulo is commonly used when looking for the remainder.
for example, 15= 1 mod 7. This means that 15 is congruent to 1 mod 7 or the remainder of 15 divided by 7 is 1. The phi function on the other hand is phi of n (1-1/n1)(1-1/n2)...Xn
or in simpler terms, 92 has factors 2^2x23. thus
phi of 92 =(1-1/2)(1-1/23) X 92. this gives 1/2X22/23X92 or 44. thus phi of 92 = 44. Substituting these values,
19^phi of 92 = 1 mod 92
=> 19^44 = 1 mod 92
we can square both sides yielding:
(its a property of modulo)
=>19^88 = 1^2 mod 92
for the finishing step, multiply 19^4 to both sides to obtain 19^92.
thus, 19^92 = 19^4 mod 92
19^4 = 130321
so, 19^92 = 130321 mod 92
but, we can decrease 130321 by subtracting 92n from it where n is an integer and the result must be the least positive as possible. Thus, 130321/92 is approximately 1416. so, 92x1416 = 130272 => 130321-130272 = 49
therefore, the remainder is 49

Answer 2

Let's use some elementary number theory.
Recall that Euler's function, φ(n), is the number
of whole numbers less than n and relatively prime to n.
Also φ(ab) = φ(a)*φ(b) if a and b are relatively prime.
Finally, we have Euler's theorem:
If (a,n) = 1 then a^φ(n) = 1(mod n).
So we have
φ(92) = φ(23*4) = φ(23)*φ(4) = 22*2 = 44.
Thus
19 ^44 = 1(mod 92).
Which gives us
19^88 = 1(mod 92).
So 19^92 = 19^88 *19^4 = 19^4(mod 92).
Now 19^2 = 361 = -7(mod 92)
and
19^4 = (-7)² = 49(mod 92).
The answer is 49.

Answer 3

ok take 19 square = 361
the reminder of 361/92 is 85
so we have now to find the reminder of 85^46 over 92 (92/2 =46)
again 85 square =7225
the reminder of 7225/92 is 49
so, now we have to find the reminder of 49^23/92
we can break this number to
49^5 * 49^5 *49^5 * 49^5 *49^3/92
= 13* 13* 13* 13* 73
= 2084953
the reminder of this number over 92 is 49
The final answer is 49

Answer 4

the farthest i could reach...

(19^92)/92
= {(23-4)^92}/(23*4)

this must be a standard form... need to look for its soln

other answerers... pls take this forward...