Which Two Statements Are Related?

Number Theory Level 3

$\begin{array}{|c|} \hline x \text{ is divisible by 2.} \\ x \text{ is divisible by 3.} \\ x \text{ is divisible by 4.} \\ \hline \end{array}$

If exactly one of the statements above is true, which of these statements is definitely false?

x \text{ is divisible by 3.}

x \text{ is divisible by 4.}

x \text{ is divisible by 2.}

2 solutions

Zee Ell
Sep 7, 2016

If x is divisible by 4, then it is divisible by 2 as well (since 2 is a factor of 4), meaning that two statements would be true instead of one.

Hence, the statement which is definitely false:

$\boxed { \text { x is divisible by 4. } }$

For the sake of completeness, here are some examples (values of x), when exactly one of the other two statements are true:

$\text { x is divisible by 2: } x \in (2, 10, 14, 22, ... ); x = 12k + 2 \text { or } x = 12m + 10$
$\text { x is divisible by 3: } x \in (3, 9, 15, 21, ... ); x = 6n - 3$

Chase Marangu
May 24, 2018

If x is divisible by 4, then x is also divisible by 2, therefore x cannot be divisible by 4 with only one of the statements being true. If x is divisible by 3, then x is not divisible by 2 or 4, but if x is divisible by 2 and exactly one of the statements is true, then x is cannot divisible by 4 or 3, therefore, regardless of whether x is divisible by 2 or 3, x will not be divisible by 4.

Which Two Statements Are Related?

2 solutions

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