Help to solve $15x \equiv 20 \pmod{88}$

Question

Solve $15x$ "congruent to" $20\mod 88$

So far I think I know $15\mod 88$ is $-41$ or if positive $47$`

$15\cdot 6 = 90 \equiv 2\pmod{88} \implies 15\cdot 60 \equiv 20 \pmod{88}$ — Doug M, Dec 09 '16 at 00:08

score 1 · Answer 1 · answered Dec 08 '16 at 23:31

A general approach for problems of this type:

Notice that, if you can find an $y$ such that $15y \equiv 1 \pmod{88}$, then we can multiply both sides of this congruence by $y$ to yield $15xy \equiv 20y \pmod{88}$. This becomes $x \equiv 20y \pmod{88}$.

So how do we find this $y$? Notice that $\gcd(15, 88) = 1$. Therefore, we can use the extended Euclidean algorithm to find the guaranteed $m, n \in \mathbb{Z}$ such that $15m + 88n = 1$. Modding out by $88$ here yields $15m \equiv 1 \pmod{88}$, and so the $m$ we found is our $y$!

score 1 · Answer 2 · edited Apr 13 '17 at 12:20

1

${\rm mod}\ 88\!:\,\ x \equiv \dfrac{20}{15}\equiv \dfrac{4}3\equiv \dfrac{-84}3\equiv -28\equiv 60$

Beware $\ $ Modular fraction arithmetic is well-defined only for fractions with denominator coprime to the modulus, and we can only cancel factors coprime to the modulus (as we did for the factor $5$ above). See here for further discussion.

edited Apr 13 '17 at 12:20

Community

1

answered Dec 08 '16 at 23:40

Bill Dubuque

272,048

I think you pretty much said that, not only at the link you provided, but also in an answer at the question of which this question is a duplicate. We need folks like you to note this sort of duplicity, to not repeat an answer you've already given, or at the very least, to initiate a close as a duplicate. I have seen you do that and it's really been helpful. On the other hand, it's entirely understandable that you may have remembered it, but searching for it would have taken more time than simply answering again. – amWhy Dec 09 '16 at 00:29
This is as true for me as it is for anyone who's been 'round here for awhile. – amWhy Dec 09 '16 at 00:35
Thanks guys. Sorry to the jobsworths that moaned about me answering instead of commenting. It was my first ever post and I was unaware of the commenting process and that it would cause so much hoohar sheesh. – khaya Dec 13 '16 at 02:13

score 0 · Accepted Answer · answered Dec 08 '16 at 23:25

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A trick to remember.

You are trying to solve the equation $15x - 20 = 88y$ for some $x,y$. Factorize five out, and you get that $5(3x-4) = 88y$. Now, since $5$ is prime and $5$ doesn 't divide $88$, it divides $y$.

Now, it is a simple question of trial and error with multiples of $5$ (you don't have to check all values of $y\ $!). As it turns out, with $y=10$, $x=60$ works.

answered Dec 08 '16 at 23:25

Sarvesh Ravichandran Iyer

76,105

So multiplicative inverse of 15 modulo 88 is 47? And solving 15x =20(mod88) is x=6? – khaya Dec 08 '16 at 23:32
nice work bro . – wesssg Dec 08 '16 at 23:32
The solution to "$15x$ congruent to $20 \mod 88$" is $x=60$. The multiplicative inverse is $47$, as you have said. Also, @wesssg thank you. – Sarvesh Ravichandran Iyer Dec 08 '16 at 23:35

Help to solve $15x \equiv 20 \pmod{88}$

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