How many positive two-digit integers are both multiple of 8 and divisible by 6?

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Illvoices

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How many positive two-digit integers are both multiple of 8 and divisible by 6?

I need to know a n easy way to solve this question:eek:
 
Since you've shown no work of your own, I'm forced to assume you have none to share and are stuck at the very beginning. I think a great place to start is to simply list all two-digit multiples of 8 and all two-digit multiples of 6, like so:

6: 12, 18, 24, ..., 90, 96
8: 16, 24, 32, ..., 88, 96

Now circle all the numbers these two lists have in common. Just from the small portion I've done for you, you can already see that 24 and 96 are common to both lists. Are there any other numbers? What do these numbers have in common? Now for the real test of understanding: Suppose you were asked the same exercise but for two-digit multiples of 5 and 7. What would the answer be then? You can repeat the list exercise if you need. What do you notice about these numbers? How does this all generalize to help you find common multiples of any arbitrary x and y? As a hint, consider the process you would take to find the answer to 1/6 + 1/8 or 1/5 + 1/7. How does that process relate to finding common multiples?
 
The prime factors of 6 are 2(3). A number is divisible by 6 if and only if it has those prime factors possibly with others. The prime factors of 8 are (2)(2)(2). A number is divisible by 8 if and only if it has those prime factor possibly with others. A number is divisible by both 6 and 8 if and only if it has prime factors (2)(2)(2)(3) so is a multiple of 24.
 
Illvoices said:
How many positive two-digit integers are both multiple of 8 and divisible by 6?

I need to know a n easy way to solve this question:eek:
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Use what they taught you in class, about finding Least Common Multiples (here). Then figure out how many multiples of that you can find, that have two digits (rather than three or more). ;)
 
stapel said:
Use what they taught you in class, about finding Least Common Multiples (here). Then figure out how many multiples of that you can find, that have two digits (rather than three or more). ;)
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Stapel, how are you? This particular student is studying for her SAT test so there is really nothing to look at from (current) class notes.
 
Illvoices said:
How many positive two-digit integers are both multiple of 8 and divisible by 6?

I need to know a n easy way to solve this question:eek:
Click to expand...
As Halls said, just look at multiples of 24. 24, 48, 72, 96 and 120. So the answer is 4.
 
Jomo said:
Stapel, how are you? This particular student is studying for her SAT test so there is really nothing to look at from (current) class notes.
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First, how did you know that "this particular student is studying for her SAT test"?

Second, she should have learned something in some class about this!
 
HallsofIvy said:
First, how did you know that "this particular student is studying for her SAT test"?

Second, she should have learned something in some class about this!
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This particular student has been asking a number of questions and in one of the earlier posts mentioned preparing for the SAT exam. As far as what she learned in her prior classes, I think that in this case we have a good opportunity to show the quickest/best way we would do the problems vs doing it the way the teacher wants. Also, I believe that these are SAT type problems just because of the type of problems.
 
Yes I am studying for the SAT and I plan on taking it on June or Aug. I haven't been practicing my math that much and I thought the site was a great way to boost my score. Oh and one more thing I'm a male boy not female ;)
 
I've got a question about the L CM of 6 and 8. How can I know the following least common multiple after solving for 2*2*2*3. How do I keep adding factors of the same multiple to make it known for the next.
 
Illvoices said:
I've got a question about the L CM of 6 and 8. How can I know the following least common multiple after solving for 2*2*2*3. How do I keep adding factors of the same multiple to make it known for the next.
Click to expand...
I truly am not sure I understand this question at all.

LCM stands for LEAST common multiple so it makes no sense to talk about the next one when there is only one.

Let's start by defining a multiple of the integer a. We define b as a multiple of a if and only if there exists an integer c such that

a∗c=b\displaystyle a * c = b.

The integer a has an infinite number of multiples, such as 2b, 3b, 4b and so on as well as 0, - b, - 2b, - 3b. With me so far?

We define b as the least multiple of a if and only if a is a positive integer and b is the smallest positive integer that is a multiple of a. The least multiple is unique.

Got the distinction between a multiple and a least multiple?

The word "common" in "common multiple" is from an old usage of "common" that means "shared." We define u as a common multiple of distinct integers m and p if and only if there exists a pair of distinct integers n and q such that

m∗n=u=p∗q.\displaystyle m * n = u = p * q..

Integers m and p have an infinite number of common multiples, namely u times any integer.

Still with me?

We define u as the least common multiple of m and p if and only if m and p are positive integers and u is the smallest positive integer that is a least common multiple.

The least common multiple is unique.

Got that step?

The quickest way to find a common multiple of m and p is to multiply m and p together. What we have done is match the pair m and p with the pair p and m so that

m∗p=v=p∗m.\displaystyle m * p = v = p * m. Clearly v is a common multiple of m and p.

Although that process gives a common multiple, it is not necessarily the least common multiple. To find the least common multiple, you must first identify the prime divisors of both numbers.

Now how what is your question?
 
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Illvoices said:
I've got a question about the L CM of 6 and 8. How can I know the following least common multiple after solving for 2*2*2*3. How do I keep adding factors of the same multiple to make it known for the next.
Click to expand...
You don't mean "the following least common multiple", you mean "the next multiple". And you get that by multiplying by the smallest multiple, 2, to get 2(24)= 48. The next smallest number is 3 and 3(24)= 72. 4(24)= 96 and the next multiple 5(24) is larger than 100. The "common multiples of 6 and 8" are 23, 48, 72, and 96 as I said before.
 
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