Math puzzle

[JMF]

From high school days, some genuine recreational math:

A math professor was talking with his assistant.

"Yesterday I met three people," said the professor. "When you multiply
their ages together, you get 2450. When you add their ages together, it
adds up to twice your age. What are their ages?"

The assistant shrugged his shoulders and said, "I don't know."

The professor said, "By the way, I'm older than all of them."

The assistant then told the professor the ages of the three people.

Question for the reader: what are the ages of the professor, the
assistant, and the three people?

[Frank J. Lhota]

On 4/24/2013 8:32 AM, JMF wrote:
> From high school days, some genuine recreational math:
>
> A math professor was talking with his assistant.
>
> "Yesterday I met three people," said the professor. "When you multiply
> their ages together, you get 2450. When you add their ages together, it
> adds up to twice your age. What are their ages?"
>
> The assistant shrugged his shoulders and said, "I don't know."

The number 2450 factor into the primes 2 * 5 * 5 * 7 * 7, so the prime
factors from the three peoples ages must be these 5 primes. Now we do
not know the age of the assistant, but certainly the assistant does. If
there were only one factorization of 2450 into three ages that adds up
to twice his age, then he would know their ages. But he didn't, so there
must be 2 factorizations of 2450 into 3 ages that have the same sum.
There is only such one pair of factorizations:

5 + 10 + 49 = 7 + 7 + 50 = 64

From this, we can deduce that the assistant is 32 years old.

> The professor said, "By the way, I'm older than all of them."
>
> The assistant then told the professor the ages of the three people.
>
> Question for the reader: what are the ages of the professor, the
> assistant, and the three people?

The professor clearly must be at least 49 years old. If she were older
than 50, then saying "I'm older than all of them" still leaves two
possibilities for the three people's ages. But he was able to tell her
their ages, so she must be 50 years young, and the three people's ages
are 5, 10, and 49.

--
"All things extant in this world,
Gods of Heaven, gods of Earth,
Let everything be as it should be;
Thus shall it be!"
- Magical chant from "Magical Shopping Arcade Abenobashi"

"Drizzle, Drazzle, Drozzle, Drome,
Time for this one to come home!"
- Mr. Wizard from "Tooter Turtle"

[Frank J. Lhota]

On 4/24/2013 9:06 PM, Frank J. Lhota wrote:
> On 4/24/2013 8:32 AM, JMF wrote:
>> From high school days, some genuine recreational math:
>>
>> A math professor was talking with his assistant.

Sorry, I missed that line first time around and assumed the professor
was female.

[JMF]

On 4/25/2013 03:06, Frank J. Lhota wrote:
> On 4/24/2013 8:32 AM, JMF wrote:
>> From high school days, some genuine recreational math:
>>
>> A math professor was talking with his assistant.
>>
>> "Yesterday I met three people," said the professor. "When you multiply
>> their ages together, you get 2450. When you add their ages together, it
>> adds up to twice your age. What are their ages?"
>>
>> The assistant shrugged his shoulders and said, "I don't know."
>
> The number 2450 factor into the primes 2 * 5 * 5 * 7 * 7, so the prime

> factors ...

Well done indeed, perfect analysis.