Math Help Forum: 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.

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.

im not sure i follow you

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 . . .







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
. . is divisible by 12 without using a calculator or doing long division.
. . Explain your reasoning.

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

Since ,
. . which is divisible by 3, then is divisible by 3.


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

Since ends in , which is divisible by 4, then is divisible by 4.


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

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.

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 and then (and when and are coprime we have ).