Factors of Integers

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Jand

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Here is a statement that was marked as true: "A number is a multiple of 45 if and only if it is divisible by both 5 and 9."

How is this so? The factors of 45 are 1x45, 3x15, and 5x9.
The prime factors of 45 are 3^2x5.

I'm baffled. Why is 3 and 15 excluded from the answer (obviously factors of (5 AND 9) exclude 15 as a factor, and factors of (5 AND 9) exclude 3 as a factor). What am I missing? How was (5 AND 9) discovered to show that a number that is a multiple of 45 must be divisible by BOTH 5 and 9?
 
Jand said:
Here is a statement that was marked as true: "A number is a multiple of 45 if and only if it is divisible by both 5 and 9."
Why is 3 and 15 excluded from the answer (obviously factors of (5 AND 9) exclude 15 as a factor, and factors of (5 AND 9) exclude 3 as a factor). What am I missing? How was (5 AND 9) discovered to show that a number that is a multiple of 45 must be divisible by BOTH 5 and 9?
Click to expand...

3 & 15 are not excluded. They are not under consideration. Only 5 & 9 are.

If X is divisible by 45 then it is divisible by 5 & 9.

If X is divisible by 5 & 9 then it is divisible by 45.

That is all the if & only if says.
 
pka said:
3 & 15 are not excluded. ...
Click to expand...

Just to put it another way (and offer a hint on the solution to the problem):

The statement "obviously factors of (5 AND 9) exclude 15 as a factor, and factors of (5 AND 9) exclude 3 as a factor)" is incorrect:
If n is divisible by 5 and 9 then n is a multiple of 9, i.e. n = 9 m1. It is also a multiple of 3 since n=3*(3*m1). Since 3 and 5 are relatively prime, n is also a multiply of 3 and 5, i.e. n = 3 * 5 * m2 = 15 m2. Thus it is a multiple of 15.
 
Ishuda said:
Just to put it another way (and offer a hint on the solution to the problem):

The statement "obviously factors of (5 AND 9) exclude 15 as a factor, and factors of (5 AND 9) exclude 3 as a factor)" is incorrect:
If n is divisible by 5 and 9 then n is a multiple of 9, i.e. n = 9 m1. It is also a multiple of 3 since n=3*(3*m1). Since 3 and 5 are relatively prime, n is also a multiply of 3 and 5, i.e. n = 3 * 5 * m2 = 15 m2. Thus it is a multiple of 15.
Click to expand...
@Ishuda, Do you know what if and only if means in mathematics?
The original question is about 5, 9, & 45. There is absolutely no word about 3 nor 15. Neither of those two has anything whatsoever to do with the question.

Again the statement is:
A number is divisible by 45 if and only if it is divisible by both of 5 & 9.
 
Yes, Ishuda clearly knows what "if and only if" means! Perhaps the problem is that you are reading more into the words than they actually mean.

"A number is divisible by 45 if and only if it is divisible by 5 and 9."

If a number is divisible by 45 then it is of the form 45k for some integer k. 45k= 5(9k) and 45k= 9(5k) so it is clearly divisible by 5 and 9.
"If a number is divisible by 45 then it is divisible by 5 and 9."

If a number is divisible by 5 then it is of the form 5k for some integer k. It that number is also divisible by 9, then, since 5 is prime and not divisible by 9, k must be of the form 9n for some integer n. That is, 5k= 5(9n)= 45n so the number is divisible by 45.
"If a number is divisible by 5 and 9 then it is divisible by 45." which is the same as
"A number is divisible by 45 only if it is divisible by 5 and 9."

Therefore: "A number is divisible by 45 if and only if it is divisible by 5 and 9".
 
HallsofIvy said:
Yes, Ishuda clearly knows what "if and only if" means! Perhaps the problem is that you are reading more into the words than they actually mean.

"A number is divisible by 45 if and only if it is divisible by 5 and 9."

If a number is divisible by 45 then it is of the form 45k for some integer k. 45k= 5(9k) and 45k= 9(5k) so it is clearly divisible by 5 and 9.
"If a number is divisible by 45 then it is divisible by 5 and 9."

If a number is divisible by 5 then it is of the form 5k for some integer k. It that number is also divisible by 9, then, since 5 is prime and not divisible by 9, k must be of the form 9n for some integer n. That is, 5k= 5(9n)= 45n so the number is divisible by 45.
"If a number is divisible by 5 and 9 then it is divisible by 45." which is the same as
"A number is divisible by 45 only if it is divisible by 5 and 9."

Therefore: "A number is divisible by 45 if and only if it is divisible by 5 and 9".
Click to expand...
@HallsofIvy, Did you read the entire thread If so, to whom is this reply addressed?
 
pka said:
@Ishuda, Do you know what if and only if means in mathematics?
The original question is about 5, 9, & 45. There is absolutely no word about 3 nor 15. Neither of those two has anything whatsoever to do with the question.

Again the statement is:
A number is divisible by 45 if and only if it is divisible by both of 5 & 9.
Click to expand...
pka,
Sorry if I confused you. Yes I know what if and only if means. My post
Just to put it another way (and offer a hint on the solution to the problem):

The statement "obviously factors of (5 AND 9) exclude 15 as a factor, and factors of (5 AND 9) exclude 3 as a factor)" is incorrect:...
Click to expand...

was an agreement with what you said about the exclusion of 15 and 3 as factors and, hopefully a hint as to how to solve the problem. Since you apparently didn't understand what I said, I will explain more fully:

The "Since 3 and 5 are relatively prime" was supposed to get the original poster to consider the fact that 5 and 9 are relatively prime and the consequences of that similar to the n = 15 m2 (after correcting the typo), i.e. n=45*m since 5 and 9 are relatively prime.

The other side of the coin [n a multiple of 45] was demonstrated by the n = 9 m1 = 3 (3 * m1) in the same fashion as n=45m = 5*(9m)=9*(5m) showing n was divisible by 9 and 5.

Thus both ways were proved and it is an "if and only if"
 
I think I understand. Thanks for the help.

The first integer on the number line that divides by (both 9 AND 5) is 45.

So the "if and only if stuff" means:

If (an integer divides by 45) then (the integer divides by 5 AND 9) - both true or neither true

I need to take more care and time to understand the precise word language of mathematics that accompanies formulas and numbers as I learn. I was assuming things that were not even mentioned in the statement.
 
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