Assume you don't participate in gyms

Number Theory Level 3

I've been instructed to hatch some eggs. For every egg that I hatch, I will receive either 700 or 1700 stardusts.

Would I ever be able to accumulate precisely 8800 stardusts?

Note: Stardust is the in-game currency in Pokémon Go.

Yes No

1 solution

Shourya Pandey
May 11, 2017

Solution 1 : By the Chicken McNugget Theorem (yes, that is the name),

$7\times 17 - 7 - 17 = 95$ cannot be represented as a linear combination of 7s and 17s.

Now if $88$ could be represented in that form, then so could $95=88+7$ , a contradiction. The answer to the question is $\boxed{No}$ .

Solution 2 : Assume the contrary. Then for some non-negative integers $a$ and $b$ , we have

$700a + 1700b = 8800$ , or

$7a+17b = 88$ .

Taking modulo $7$ , we get $3b \equiv 4 \pmod{7} \equiv 18 \pmod{ 7}$ , so

$b \equiv 6 \pmod{7}$ , so $b \geq 6$ .

But then $7a+17b \geq 17\times 6 = 102 > 88$ , a contradiction. Hence our assumption is incorrect, and the answer to the question is $\boxed{No}$ .

Thanks for the solution!

By the way, there's a one-line (or two-line solution).

Pi Han Goh - 4 years ago

I added another solution. Was that what you were looking for?

Shourya Pandey - 4 years ago

Yes yes! Great!!!

Pi Han Goh - 4 years ago