Divisibility Question?

Question

Hello,

Is is true that, if a number n is divisible by a, by b, and by c, then it is also divisible by a*b*c? Why?

Thank you

I would also like to see a proof,

thank you

Answer 1

Only if a, b and c are relatively prime.

e.g.1
Consider the case of 3,4,5 whose product = 60.
Any number divisible by 3, 4 and 5 will also be divisible by 60.

e.g.2
Consider the case of 3,5,6 whose product = 90.
You can verify that 30 and 60 are divisible by 3, 5 and 6 but not 90. That's because 3 and 6 are not relatively prime (they have common factors).

*EDIT*
Proof of what? The statement is only conditionally true.

Answer 2

You can divide any number by any number except for zero. Since you didn't mention any restrictions on the answer is as stated true.

If you want to restrict the answer to say whole numbers, which I'm betting is the real question you want answered, then the answer is false. I have provide two proofs both are relatively easy.

A simple way to disprove it is to have a,b, and c, such that when multiplied they are greater than the number n, the result would be less than one and greater than zero every time.

Example 1:
n = 18
a = 9
b = 6
c = 3
n/a = 18/9 = 2
n/b = 18/6 = 3
n/c = 18/3 = 6
n/abc = 18/36 = 0.5

Even easier let a = n. Therefore n/a = 1, if either b or c is greater than one you have a fraction between zero and one as the result.

Example 2:
n = 6
a = 6
b = 3
c = 1
n/a = 6/6 = 1
n/b = 6/3 = 2
n/c = 6/1 = 6
n/abc = 6/18 = 0.33333

Answer 3

Example 1
n = 36
a = 4
b = 12
c = 18

n/abc = 36/4 x 12 x 18 = 36/864
---------------

Example 2
n = 72
a = 24
b = 12
c = 6

n/abc = 72/24 x 12 x 6 = 72/288
---------------

Example 3
n = 48
a = 2
b = 4
c = 6

n/abc = 48/2 x 4 x 6 = 48/48 = 1
---------------

∴example 1 & 2 show that n CANNOT be divisible by abc(a*b*c),but example 3 shows that n CAN be divisible by abc(a*b*c).Therefore,n can be divisible by abc SOMETIMES.

Answer 4

n is 24

a is 3
b is 2
c is 4

24 divided by 9 doesn't equal a whole number

Answer 5

It's true.