divisibility test

[Guest]

pLease help me ;

decide wether the following are true of false using only divisibility test, give reason for the answer:

24 / 325,608

45 / 13, 075

40/ 1,732,800

36 / 677,916

Prove in two different ways that 2 divides 114.

Prove in two different ways that 3 /336

Also if 24 divides b, what else must divide b?[/tex][/code]

 

[stapel]

You ask for the truth or falsity of the expressions, but I'm not sure what that means. I mean, "24 ÷ 325,608" is just a number; I'm not sure what "truth" or "falsity" would mean in such a context...? Or do you mean the "slash" to stand for "divides into" (in a number-theory sort of way), rather than the usual "divided by" meaning...? So "24 / 325,608" means "twenty-five is a divisor of 325,608"...?

Thank you.

Eliz.

 

[soroban]

Hello, bware!

Are you aware of the various divisibility tests?

Decide wether the following are true of false using only divisibility test, give reason for the answer:

$\displaystyle 24\;|\;325,608$

$\displaystyle N\,=\,325,608$
The sum of the digits of $\displaystyle N$ is divisible by 3 $\displaystyle \,(3\,+\,2\,+\,5\,+\,6\,+\,0\,+\,8\:=\:24)$
$\displaystyle \;\;$ Hence, $\displaystyle N$ is divisible by 3.

The last three-digit number of $\displaystyle N$ is divisible by 8 $\displaystyle \,(608\,\div\,8\:=\:76)$
$\displaystyle \;\;$Hence, $\displaystyle N$ is divisible by 8.

Therefore: $\displaystyle N$ is divisible by: $\displaystyle 3\cdot8\:=\:24$

$\displaystyle 45\,|\,13,075$

The number ends in 5. $\displaystyle \;$Hence, $\displaystyle N$ is divisible by 5.

But the sum of the digits, $\displaystyle 1\,+\,3\,+\,0\,+\,7\,+\,5\:=\:16$, is not divisible by 9.

Therefore: $\displaystyle N$ is not divisible by 45.

$\displaystyle 40\,|,1,732,800$

Since $\displaystyle N$ ends in 0, it is divisible by 5.
Since the last three-digit number is divislble by 8 $\displaystyle \,(800 \,\div\,8\:=\:100)$, it is divisible by 8.

Therefore, $\displaystyle N$ is divisible by: $\displaystyle 5\cdot8\:=\:40$

$\displaystyle 36\,|\,677,916$

The sum of the digits $\displaystyle \,(6\,+\,7\,+\,7\,+\,9\,+\,1\,+\,6\:=\:36)$ is divisible by 9.
$\displaystyle \;\;$Hence, $\displaystyle N$ is divisible by 9.
The last two-digit number is divisible by 4 $\displaystyle \,(16\,\div\,4\:=\:4)$.
$\displaystyle \;\;$Hence, $\displaystyle N$ is divisible by 4.

Therefore, $\displaystyle N$ is divisible by: $\displaystyle 9\cdot4\:=\;36$