A number theory problem by Nikola Alfredi.

Number Theory Level pending

A function is defined as $\displaystyle f(x,y) = 23 \times 34^{x} + 5590 y + 588$ .

Given that $\displaystyle \exist {x,y} \in \mathbb{Z}^{ +} \ \ : f(x,y) \equiv 0 \pmod {43}$ .

If $\displaystyle f(x,y) \equiv N \pmod {559}$ . Then what is the value of $N$ . If you get multiple values of $N$ , then submit the answer as the sum of all the values $N$ .

The answer is 344.

1 solution

A Former Brilliant Member
Mar 13, 2020

$5590y+559\equiv 0\pmod {43}$ for all integer $y$ . So we have to find some positive integer $x$ for which $23\times 34^x+29\equiv 0\pmod {43}$ holds. By direct inspection we see that $x=2$ is a candidate for this. For $x=2, 23\times 34^x+29=26617\equiv \boxed {344}\pmod {559}$ .

This is same, how I did.

Nikola Alfredi - 1 year, 3 months ago

Did you check for the other possible values of $x$ ? Is it true that the result holds for all values of $x$ satisfying the congruence modulo $43$ ?

A Former Brilliant Member - 1 year, 3 months ago

I checked for quite few values ... then looked for its' multiples and came out with the result that $x = 42k + 2 \ \ \ \forall k \in \mathbb{Z}^{\geq 0}$ .

Nikola Alfredi - 1 year, 3 months ago

And $\displaystyle f(x,y) \equiv 0, 344 \pmod {559} \ \ \ \forall {x,y} \in \mathbb{Z} ^{+}$ .

Nikola Alfredi - 1 year, 3 months ago

@Alak Bhattacharya [Check this out.](https://brilliant.org/problems/5-card-combintions-watch-out/?ref_id=1585091)

Nikola Alfredi - 1 year, 3 months ago

What is the connection between this question and that I didn't get.

A Former Brilliant Member - 1 year, 3 months ago

No connection, It is a fun question... It feels satisfying after finding the answer.

Nikola Alfredi - 1 year, 3 months ago

A number theory problem by Nikola Alfredi.

The answer is 344.

1 solution

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