if 6^y is a factor of (2^14)*(3^24)

[larksleep]

Hi: Could someone please very simply explain how to go about solving this? Many Thanks!!!

if 6^y is a factor of (2^14)*(3^24) , what is the greatest possible value for y?

a)7
b)8
c)14
d)18
e)36

 

[Ishuda]

Hi: Could someone please very simply explain how to go about solving this? Many Thanks!!!

if 6^y is a factor of (2^14)*(3^24) , what is the greatest possible value for y?

a)7
b)8
c)14
d)18
e)36

Since you have a 6 to some power as a factor, you would like to see a 6 to some power in the number. Well 2¹⁴ 3²⁴ = 2¹⁴ 3¹⁴ 3¹⁰ = ?. Hint: a^x b^x = (a b)^x

 

[larksleep]

Since you have a 6 to some power as a factor, you would like to see a 6 to some power in the number. Well 2¹⁴ 3²⁴ = 2¹⁴ 3¹⁴ 3¹⁰ = ?. Hint: a^x b^x = (a b)^x

Thanks Ishuda. I'm getting that the answer is 14, but I'm a little hesitant on the process i'm taking. Could you (or someone else) write a similar level problem and I'll see if I can get the answer correct?

Thanks for your help!

 

[Ishuda]

Thanks Ishuda. I'm getting that the answer is 14, but I'm a little hesitant on the process i'm taking. Could you (or someone else) write a similar level problem and I'll see if I can get the answer correct?

Thanks for your help!

The answer is 14. For a different but similar problem:
If 30^y is a factor of (2²² 3²⁴ 5⁷¹) [or maybe (4¹¹ 3²⁴ 5⁷¹)], what is the greatest possible value for y?

The idea is to make the number given into a product of prime factors, and break your number down into prime factors to see what you need out of that list.

 

[larksleep]

The answer is 14. For a different but similar problem:
If 30^y is a factor of (2²² 3²⁴ 5⁷¹) [or maybe (4¹¹ 3²⁴ 5⁷¹)], what is the greatest possible value for y?

The idea is to make the number given into a product of prime factors, and break your number down into prime factors to see what you need out of that list.

Thanks for doing this Ishuda.

Would that be y=22?

(2^Y 3^Y 5^Y)

Is there a basic rule here that you use the lowest exponent regardless of the base?

 

[Ishuda]

Thanks for doing this Ishuda.

Would that be y=22?

(2^Y 3^Y 5^Y)

Is there a basic rule here that you use the lowest exponent regardless of the base?

The answer is y=22. Yes, the lowest appropriate exponent.

Doing another problem as an example
If 90^y is a factor of (4²² 3⁴⁵ 5⁷¹ 7⁴), what is the greatest possible value for y?

First turn the given number into prime values or
(4²² 3⁴⁵ 5⁷¹ 7⁴) = (2⁴⁴ 3⁴⁵ 5⁷¹ 7⁴)

Now the number we are working with
90 = 2 3² 5
Since we have an exponent larger than 1, turn that into a single number and go back to our given number and do the same (as much as can be done).

So
90 = 2 9 5
and
(4²² 3⁴⁵ 5⁷¹ 7⁴) = (2⁴⁴ 3⁴⁵ 5⁷¹ 7⁴) = (2⁴⁴ 3 9²² 5⁷¹ 7⁴)
Pick the smallest exponent from those which are in our number, i.e. 22.

The greatest value for y is 22.

 

[larksleep]

Awesome. I think i got it. Many thanks to you, Ishuda, for taking the time to explain it!