# Find the base system

In an unknown base system two numbers are written as 402 and 302 in the base 9 system the product of these two numbers is 75583. Find the unknown base.

• This should be anwsered without – Meadara Feb 18 '16 at 22:10

## 2 Answers

$75583$ in base $9$ is $50052=2^2*3*43*97$ in base 10

The unknown base $b$ is at least $5$ since $402$ start with a $4$

If $b>10$, $402_b*302_2 \geq (11*11*4+2)*(11*11*3+2)=177390 > 50052$ so $b\leq 10$

$402_b =2 \mod(b)$ and $302_b=2 \mod(b)$ so $75583_9=50052_{10}=4\mod(b)$

$$50052=2\mod(5)$$ $$50052=0\mod(6)$$ $$50052=2\mod(7)$$ $$50052=4\mod(8)$$ $$50052=3\mod(9)$$ $$50052=2\mod(10)$$

So $b=8$

• Thanks for the feedback, i just want to ask why is 402 to the base b equal to 2 mod b. – Meadara Feb 18 '16 at 22:27
• @Meadara because $402_b=4*b*b+0*b+2$ – stity Feb 18 '16 at 23:58
• I see, even though it is not obvious at first sight (for me) – Meadara Feb 19 '16 at 0:08

HINT.

You must solve the equation $(4x^2+2)(3x^2+2)=50052$, where $50052=(75583)_9=2^2\cdot3\cdot43\cdot97$.

• Great solution but it doesn't work if the second digit is not $0$ because you would get some $x^3$ and $x$ – stity Feb 18 '16 at 22:14
• Thats a bit too straaight foward and this question was made to solve without a calculator. – Meadara Feb 18 '16 at 22:20
• @stity this works more generally and is very simple, since the equation must have an integer root. – Mathmo123 Feb 18 '16 at 22:33