What do I choose?

Number Theory Level 2

"Very well, you have completed my first puzzle. Now, I will give you two choices.

Option 1. I will randomly choose a number, $n$ , from 1-100 inclusive. If the following statement: $3|5n^2+7n-4$ is true, you will live, or else, the walls will close in and crush you.

Option 2. I will roll a 6 sided die, if the resulted number is divisible by 3, you will live, else, you know what happens.

Which choice would have a higher probability of allowing you to live?

It doesn't matter Option 2 Impossible to tell Option 1

1 solution

Satvik Golechha
Jan 5, 2015

Option 1

$n$ is chosen from $1$ to $100$ .

$3|5n^2+7n-4$ and $3|3(2n^2+3n-1)$ . Subtracting the first from the second, we have $3|(n+1)^2$ .

Since $3$ is a prime, we must have $3|n+1$ . That is, $n=3k-1$ for natural $k$ . From $1$ to $99$ , we have $\dfrac{1}{3}$ probability and it reduces a bit (let's say x)after $100$ since $100$ doesn't follow it.

Finally, probability is $\dfrac{1}{3}-x$ .

Option 2

Only $3$ and $6$ out of $(1,2,3,4,5,6)$ follow the condition. Hence probability is $\dfrac{1}{3}$ .

Since $x>0$ , Option $2$ has a higher probability.

Option 1, pretty obviously, has probability 0.33. Also, though I like the way you tackled Option 1, I found solving $5n^{2}+7n-4 \equiv 2n^{2}+4n+2 \equiv 2(n+1)^{2} \equiv 0 \pmod{3}$ to be slightly more intuitive.

Jake Lai - 6 years, 5 months ago

What do I choose?

1 solution

Option 1

Option 2

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