A simple calculator – 3

Number Theory Level 3

A simple calculator only has 2 functions.

It always starts at 0 and then does one of the following

add $a \in \mathbb{N}$ to the number
multiply the number by $b \in \mathbb{N}$

It can do multiple operations in a row, the result of the last operation is always the input for the next one.

This calculator is then named $(a,b)$ .

Now, we introduce a new function:

raise the number to the power of $c \in \mathbb{N}$

Calculators that can do this are called $(a,b,c)$ .

Consider the calculator $(20,19)$ and the calculator $(20,19,2)$ .

Are there any positive integers that $(20,19,2)$ can calculate, but $(20,19)$ can't?

Yes No

1 solution

A Former Brilliant Member
Dec 26, 2018

We only need to know if for all $A^2=A.A$ , where $A$ is made by $(20,19)$ , there is a way to make $A^2$ using only $(20,19)$ .

There is one of the following possibilities: $A=A_0+20$ or $A=A_0.19$ .

1- for the first case: $A^2=2.20.A_0+20^2+A_0^2$

2- for the second case: $A^2=19^2.A_0^2$

We also know that $A_0$ is a multiples of $20$ anyways. so, in both cases, $A^2$ is a multiple of $20$ and can be made by adding enough number of $20$ .

A simple calculator – 3

1 solution

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