Find the remainder when (19)^92 is divided by 92.
Chinese Remainder theorem (along with other results). First note 92= 4 Γ 23 with gcd
(4,23) =1. Let us call N= 1992. We will compute, N(mod 4) and N (mod 23) and then use CRT to
compute N (mod 92).
First, N (mod 4) = (19)
92( πππ 4) = (β1)
92(πππ 4) = 1
and π(πππ 23) = 194
.[(19)
22 (πππ 23)]
2
(πππ 23) = (β4)
4
(πππ 23) = (16)
4
(πππ 23) = (β7)
2
(πππ 23) = 49
(πππ 23) = 3.
Note in the above we have used Fermatβs Little Theorem. Now, If you know CRT, you can
directly say π( πππ 92) = 49.
If not, you can compute it. One way to do it is write down two lists of numbers (one for each
relation) and pick out the first common number
Interesting. Do you know how to get started on this problem?
Well, dividing (19)^92 by 92 is like trying to divide a herd of clowns by a unicycleβit's a bit tricky. But fear not, for Clown Bot is here to help!
Now, let's get right to the math. To solve this, we need to find the remainder when (19)^92 is divided by 92. Now, 92 is not a prime number, so we can't use any fancy tricks like Fermat's Little Theorem or Euler's Totient Theorem. Time to unleash the power of brute force!
Warning: This may cause your brain to perform acrobatics, so prepare yourself for a wild ride.
We start by figuring out the remainders when we divide 19 raised to different powers by 92:
(19)^1 % 92 = 19
(19)^2 % 92 = 81
(19)^3 % 92 = 69
(19)^4 % 92 = 61
(19)^5 % 92 = 29
(19)^6 % 92 = 1
...
Notice a pattern? If you keep dividing the power of 19 by 92, the remainders repeat. And since 92 isn't our lucky number, it will keep cycling through these remainders.
Now, since 92 is not a factor of 19, the cycle will repeat every (92 - 1) = 91 powers of 19. So if we divide 92 by 91, the remainder would be 1.
Therefore, (19)^92 % 92 would have the same remainder as (19)^1 % 92, which is 19.
And just like that, the clown car of calculations comes to a stop, revealing the remainder when (19)^92 is divided by 92: 19!
I hope my clownful explanation helped bring a smile to your face.
To find the remainder when a number is divided by another number, we can use the concept of modular arithmetic.
In this case, we need to find the remainder when 19^92 is divided by 92.
Step 1: Break down 92 into its prime factors: 92 = 2^2 * 23.
Step 2: Determine the remainder when 19 is divided by each prime factor separately.
- Remainder when 19 is divided by 2: 19 % 2 = 1.
- Remainder when 19 is divided by 23: 19 % 23 = 19.
Step 3: Use the Chinese Remainder Theorem to find the remainder when 19^92 is divided by 92.
- First, find the remainders for each prime factor raised to the power of 92:
- Remainder when 1^92 is divided by 4: Since any number raised to the power of 92 is 1, the remainder is 1.
- Remainder when 19^92 is divided by 23: To find the remainder of 19^92 divided by 23, we need to use Euler's totient theorem. The totient of 23 is 22, so we have 19^22 β‘ 1 (mod 23). Therefore, 19^92 β‘ (19^22)^4 * 19^4 β‘ 1^4 * 19^4 β‘ 19^4 β‘ 14 (mod 23).
- Now, we can find the number that satisfies both conditions:
- By solving the system of congruences:
- x β‘ 1 (mod 4)
- x β‘ 14 (mod 23)
- Using the Chinese Remainder Theorem, we find that the solution is x β‘ 39 (mod 92).
Therefore, when (19)^92 is divided by 92, the remainder is 39.