CPhill

Well....Ms. SmartyPants......I even provided a general solution.....so.....take that  !!!   

The oldest child is 24 years older than the youngest

Let y be the time elapsed from the birth of the youngest to the time when the oldest is 5 times that age......so we have....

5y    = 24 + y     subtract y from each side

4y = 24     then y = 6

So, six years from the birth of the youngest, the oldest will be 30 and the youngest will be 6....thus.....the oldest is five times as old as the youngest at that time......

Then, the second to the youngest will be 8  at that time....two years older than the youngest....!!!

I'm going to employ maximum cheating on this one.......

Using the Euclidian algorithm, we have

1027  = 1(712) + 315

712 = 2(315) + 82

315 = 3(82) + 69

82 = 1(69) + 13

69 = 5(13) + 4

13 = 3(4) + 1

4 = 4(1) + 0

                              1        2        3         1        5          3          4

1  0                        1        2        7         9        52     165       712     

0  1                        1        3       10       13       75      238     1027

Take the  determinant of          165     712

                                                 238   1027        =   -1

So.....

1027(165) - 712(238) = - 1   but...we need a  "+1 "   .....so......multiplying through by -1, we have

1027(-165) + 712(238) = 1

So....x = -165   and y = 238....however.....this is only one solution.....the general solution is given by

x , y   =  [ (-165 + 712k), (238 - 1027k) ]

To "b" or not to "b"......that is the question......is it nobler in the minds of men to expandeth answers to infinity through the Queen's and heureka's employment of LaTex, or, should other - more compact - answers serve our  purposes just as well ????

That's pretty neat, Alan.....so......can we assume that....the colder it gets, the more time flies????

 [ Regardless of the climate change....I still gave you 3 points....  !!!!]

I like that presentation, Melody......it really shows how each component of the function changes the shape/orientation of the curve.....!!!!

Here's my attempt at (a).....

Notice that .....

500/7  = 71 numbers that have at least one factor of 7

500/49  = 10 more numbers that have an additional  factor of 7

500/343 = 1 more number that adds an additional factor of 7

So.....71 + 10 + 1   = 82

So ..... 7^82  will divide 500!....thus, n = 82

Sorry, Anonymous....I'm sticking with my answer......the question doesn't specify how many prime factors we can use (or even any restictions relating to the powers on those primes).....it just wants to know the smallest possible positive integer that has 6 perfect square divisors.....

Of course.....I suppose that we might claim that " 1 " could have as many perfect square divisors as we wished, but that answer seems trivial.......I'm assuming that all the perfect square divisors are unique....

And 144 is certainly < 518400 or 1296   !!!!

Note that 1001 factors as 7 x 11 x 13 = 7 * 143

So.....143 is the greatest proper divisor of 1001

This means that we have  143( m + n)  = 143m + 143m = a + b = 1001  .....where ( m + n) = 7  ....and since m + n are two positive integers whose sum = 7, they are relatively prime to each other

Thefore, the two integers 143m, 143n  = a, b  have only the factor of 143 in common.....and it is the gcd of both.

{Could some other mathematician check my logic, here ??? }

The integers must be 4, 5 and 7....since 4 * 5 * 7 =  140.....so the sum is 16

If there were only 2 integers, the least sum would be 24.....  (14 *10)

And there could not be 4 (or more) distinct integers since 140 factors as 2 * 2 * 5 * 7