Three numbers

Number Theory Level 3

There are three numbers on the board ( $a, b, c$ ).

Step: Delete one of the numbers. If $N$ equals to the sum of the two other numbers, then write down $N-1$ .

You can repeat the step above any number of times. Imagine that sometime there are three numbers on the board: $17,1983,1999$ .

Which of the following sentences is/are true?

A : It is possible, that the original three numbers are $(a,b,c)=(2,2,2).$

B : It is possible, that the original three numbers are $(a,b,c)=(3,3,3).$

Neither A nor B Only B Only A Both of A and B

1 solution

Áron Bán-Szabó
Jul 26, 2017

Note that $1999=17+1983-1$ . We will show that from after any step the sum of the two smallest numbers is $1$ bigger, than the biggest number.

Suppose that for the ( $a, b, c$ ) numbers $c=a+b-1$ . The previous three numbers are ( $a, b, x$ ). Between these numbers the $b$ is the largest. So $b=a+x-1\Leftrightarrow x=b-a+1$ .

$(17,1999,1983)\Rightarrow (17,1967,1983)\Rightarrow (17,1967,1951)\Rightarrow \dots \Rightarrow (17,15,31)\Rightarrow (17,15,3)\Rightarrow (13,15,3)\Rightarrow \dots \Rightarrow (5,7,3)\Rightarrow (5,3,3)$

We can get the ( $5,3,3$ ) numbers from the ( $3,3,3$ ) numbers, but we can't get it from the ( $2,2,2$ ) numbers.

Three numbers

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