1. ## divisibility

Investigate the questions in parts a and b by considering a number of examples

a. If a whole number is divisible both by 6 and by 2, is it necessarily divisible by 12?

b. If a whole number is divisible both by 3 and by 4, is it necessarily divisible by 12?

c. based on your examples, what do you think the answers to the questions in a and b should be?

d. Based on your answer to part c, determine whether 3,321,297,402,348,516 is divisible by 12 without using a calculator or doing long division. Explain your reasoning.

2. Originally Posted by ihavvaquestion
Investigate the questions in parts a and b by considering a number of examples

a. If a whole number is divisible both by 6 and by 2, is it necessarily divisible by 12?

b. If a whole number is divisible both by 3 and by 4, is it necessarily divisible by 12?

c. based on your examples, what do you think the answers to the questions in a and b should be?

d. Based on your answer to part c, determine whether 3,321,297,402,348,516 is divisible by 12 without using a calculator or doing long division. Explain your reasoning.
a. 6 is divisible by 6 and 2.

3. im not sure i follow you

4. Hello, ihavvaquestion!

(a) If a whole number is divisible both by 6 and by 2, is it necessarily divisible by 12?
No . . .

As chiph588@ pointed out, 6 is divisible by 6 and by 2,
. . but it is not divisible by 12.

b. If a whole number is divisible both by 3 and by 4, is it necessarily divisible by 12?
Yes . . .

$\begin{array}{ccc}\text{Multiples of 3:} & 3,6,9,12,15,18,21,24,27,30,33,36,\hdots \\
\text{Multiples of 4:} & 4,8,12,16,20,24,28,32,36,40,44,48,\hdots\end{array }$

$\begin{array}{ccc}\text{Multiples of 3 and 4:} & 12,24,36,48,60,72,84,\hdots
\end{array}$

If a number is multiple of 3 and a multiple of 4, it is a multiple of 12.

c. Based on your examples, what do you think
the answers to the questions in (a) and (b) should be?
Um . . . "No" and "Yes" ?

d. Based on your answer to part (c), determine whether $N = 3,\!321,\!297,\!402,\!348,\!516$
. . is divisible by 12 without using a calculator or doing long division.

A number is divisible by 3 if the sum of its digits is divisible by 3.

Since $3+3+2+1+2+9+7 +4+0+2+3+4+8+5+1+6 \,=\,60$,
. . which is divisible by 3, then $N$ is divisible by 3.

A number is divisible by 4 if its last two-digit number is divisible by 4.

Since $N$ ends in $16$, which is divisible by 4, then $N$ is divisible by 4.

Therefore, $N$ is divisible by both 3 and 4 . . . $N$ is divisible by 12.

5. In general, you can say that "if x is divisible by a and b, then x is divisible by a*b" if and only if if a and b are coprime.

6. Originally Posted by Tinyboss
In general, you can say that "if x is divisible by a and b, then x is divisible by a*b" if and only if if a and b are coprime.
More accurately, if $a | x$ and $b | x$ then $\textrm{lcm}{(a, b)} | x$ (and when $a$ and $b$ are coprime we have $\textrm{lcm}{(a, b)} = a \times b$).