There are 9 divisors for number A and 10 divisors for number B. The least common multiple of A and B is 2800. What are the two numbers?
Please provide a solution, not just a plain answer.
Please provide a solution, not just a plain answer.
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first, 2800 must be factorized to 2 x 2 x 2 x 2 x 5 x 5 x 7.
this is the slightly tricky part. firstly, an example. how do you find out how many factors a number has? lets take 12 for example. factories 12 and you get 2^2 x 3^1. now add one to each power and you get 3 & 2. multiply them and you get 6, the number of factors 12 has. let's make sure. it's factors are 1, 2, 3, 4, 6, 12, which is exactly 6. so what does this have to do with the question?
since A has 9 divisors, or 9 factors, 9 can only be factorized to 3 x 3. -1 from each one and we get 2 & 2. so the powers of A factorized must 2 & 2. do the same thing with B, factories it and you can only get 2 x 5. -1 from each one = 1 & 4. so B factorized has powers of 1 and 4. from this information we can determine that the only number that is a factor of 2800 and has a prime factor with a power of 4 must be 2. only 2 has four of them. so B can either be 2^4 x 7^1 or 2^4 x 5^1 hence 112 or 80. but A can only be 2^2 x5^2 hence giving us 100. last of all, to work out which one B actually is, test each one. the LCM of 100 and 80 is 400 so 80 cannot be B. the LCM of 112 and 100 is 2800 so A and B are 100 and 112. Seems to be a very long and intricate method when writing but quite easy and quick when you know how.
this is the slightly tricky part. firstly, an example. how do you find out how many factors a number has? lets take 12 for example. factories 12 and you get 2^2 x 3^1. now add one to each power and you get 3 & 2. multiply them and you get 6, the number of factors 12 has. let's make sure. it's factors are 1, 2, 3, 4, 6, 12, which is exactly 6. so what does this have to do with the question?
since A has 9 divisors, or 9 factors, 9 can only be factorized to 3 x 3. -1 from each one and we get 2 & 2. so the powers of A factorized must 2 & 2. do the same thing with B, factories it and you can only get 2 x 5. -1 from each one = 1 & 4. so B factorized has powers of 1 and 4. from this information we can determine that the only number that is a factor of 2800 and has a prime factor with a power of 4 must be 2. only 2 has four of them. so B can either be 2^4 x 7^1 or 2^4 x 5^1 hence 112 or 80. but A can only be 2^2 x5^2 hence giving us 100. last of all, to work out which one B actually is, test each one. the LCM of 100 and 80 is 400 so 80 cannot be B. the LCM of 112 and 100 is 2800 so A and B are 100 and 112. Seems to be a very long and intricate method when writing but quite easy and quick when you know how.