I Can't Tell The Difference!

Number Theory Level 1

How many of the following statements is/are true?

(1) : An odd number is never divisible by an even number.
(2) : An odd number is never divisible by another odd number.
(3) : An even number is never divisible by an odd number.
(4) : An even number is never divisible by another even number.

0 1 2 3 4

2 solutions

Finn C
May 19, 2016

This is pretty easy.

Statements

(1): An odd number is never divisible by an even number.

(2): An odd number is never divisible by another odd number.

(3): An even number is never divisible by an odd number

(4): An even number is never divisible by another even number.

Proving (in)correct

(1): An odd number is never divisible by an even number.

This is correct. Whether negative, or positive, there is no odd number divisible by an even number.

2): An odd number is never divisible by another odd number.

This is incorrect. If you multiply 2 odds they will always be an odd.

(3): An even number is never divisible by an odd number.

Wrong. Everything is divisible by 1 (which is odd) so there is no exception to that rule.

(4): An even number is never divisible by another even number.

4 is divisible by 2. In fact, every even number is either divisible by 2, or -2.

Therefore, Statement $\boxed{1}$ is the only correct statement.

All in all, all you have to follow are these rules. (E = Even, O = Odd)

E + O = O

E + E = E

O + O = O

E $\times$ O = E

$\frac{E}{O}$ = E

$\frac{O}{E}$ = Decimals

Moderator note:

Good counterexamples for 2, 3, 4.

However, you have not yet proven that statement 1 is correct. You have merely restated that "there is no odd number divisible by an even number", without offering a reason why that is true.

In response to Challenge Master:

Since no odd number contains a factor of a multiple of $2$ or $-2$ . Therefore, upon division by an even number, even though all other odd factors of both numbers may cancel out, we will still be left with a multiple of $2$ or $-2$ in the denominator. Thus proving that an odd number is never divisible by an even number.

Tapas Mazumdar - 4 years, 8 months ago

Akash Patalwanshi
May 19, 2016

Only statement ( 1 ) is true and others are false. For proving others are false:

As $3 | 27→$ an odd number can be divisible by another odd number.

As $3 | 6$ hence an even number can be divisible by odd number.

As every every even number are of the of form $2n$ where $n$ is Natural number. Hence ( 4 ) is false trivially.

I Can't Tell The Difference!

2 solutions

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