Explain the following

oldtryingtorenew

New member
Joined
Sep 6, 2022
Messages
4
Hi,
Can someone explain the following equation and how its been solved?
1= -14*43(mod 67)
43^-1 (mod 67) = -14
then the result is 53
 

Steven G

Elite Member
Joined
Dec 30, 2014
Messages
13,234
Which part needs explaining to you? Also, can you state the exact problem so we know what you are asked to solve for?
What you wrote is similar to asking how do you solve 7*2+4 = 2^4 +2.
 

mmm4444bot

Super Moderator
Joined
Oct 6, 2005
Messages
10,824
[imath]\;[/imath]
 

blamocur

Senior Member
Joined
Oct 30, 2021
Messages
1,510
Hi,
Can someone explain the following equation and how its been solved?
1= -14*43(mod 67)
43^-1 (mod 67) = -14
then the result is 53
Both side of the first identity are divided by 43 to get the second identity.
As for the "result", [imath]53 \equiv -14 \mod 67[/imath], although I am not sure what "result" means in this context. Following the advice in post #2 might clear up my confusion :)
 

Dr.Peterson

Elite Member
Joined
Nov 12, 2017
Messages
14,450
Hi,
Can someone explain the following equation and how its been solved?
1= -14*43(mod 67)
43^-1 (mod 67) = -14
then the result is 53
If you found this work in some source, it will be very helpful if you show us that source (ideally as a link or an image). We want to see what they say they are doing. They are definitely not solving an equation, since there is no unknown.

It appears that they are starting with a known fact, that [imath]1\equiv-14\cdot43(\text{mod }67)[/imath], which you can prove if you know what congruence mod 67 means; then they multiplied both sides by the multiplicative inverse of 43 (assuming it exists), to show that [imath]43^{-1}\equiv-14(\text{mod }67)[/imath]. (Note that the notation you show is incorrect.) Then we can guess that the "result" means that they have found that 53 is the least non-negative residue of -14 (by adding 67 to -14), and they are calling that the inverse of 43.

We don't know why they are doing this, or what you already understand, that we can use to explain it to you.
 
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