The King and the jester.

[mfgoode]

Find me a number" said the king to the jester " half of which is a square".
" how simple" said the jester"
.
"But" continued the king " one third of it must be a cube".
The jester looked serious for once.

"And finally " remarked the king "One fifth of it must be a fifth power"
The jester looked crestfallen.

"Solution tomorrow morning" finished the king "Or you must marry my widowed mother- in - law".

The jester was crushed.

But the next morning he had the answer and was thus saved from a fate worse than death.

See if you would have to marry the widowed mother - in - law overnight!

Mally

[drew]

Originally Posted by mfgoode

Find me a number" said the king to the jester " half of which is a square".
" how simple" said the jester"

number = 5^2a*3^2b*2^2c+1

"But" continued the king " one third of it must be a cube".
The jester looked serious for once.

number = 5^3d*3^3e+1*2^2f

"And finally " remarked the king "One fifth of it must be a fifth power"
The jester looked crestfallen.

number = 5^5g+1*3^5h*2⁵ⁱ

All 'a' through 'i' must be integers:

The exponent of 5 is 6*l = 1 (mod 5)

l=1

The exponent of 3 is 10*m = 1 (mod 3)

m = 1

The exponent of 2 is 15*n = 1 (mod 2)

n = 1

So, the number is 2¹⁵*3¹⁰*5⁶ = 30,233,088,000,000


That's a pretty big number.

[Uncwilly]

Originally Posted by drew

So, the number is 2¹⁵*3¹⁰*5⁶ = 30,233,088,000,000

In essence you are saying, in general form:
a[sup](b*c)[/sup] * b[sup](a*c)[/sup] * c[sup](a*b)[/sup]
and in our case:
a=2, b=3, c=5

[drew]

Originally Posted by Uncwilly

In essence you are saying, in general form:
a[sup](b*c)[/sup] * b[sup](a*c)[/sup] * c[sup](a*b)[/sup]
and in our case:
a=2, b=3, c=5

No, that's not a general solution because a * b = 1 (mod c) does not hold for every possible a, b and c. That was coincidentally (or by design) the case for this particular set of a, b and c.

A general solution would require additional coefficients in the exponents to force this condition.

Drew

[retina]

Does not 0 also give a correct answer? No? 0=2*(0/2)^2 etc.

[Wacky]

Originally Posted by retina

Does not 0 also give a correct answer?

Methinks that thou hast "zeroed in" on the answer.

[mfgoode]

Originally Posted by drew

number = 5^2a*3^2b*2^2c+1

number = 5^3d*3^3e+1*2^2f

number = 5^5g+1*3^5h*2⁵ⁱ

All 'a' through 'i' must be integers:

The exponent of 5 is 6*l = 1 (mod 5)

l=1

The exponent of 3 is 10*m = 1 (mod 3)

m = 1

The exponent of 2 is 15*n = 1 (mod 2)

n = 1

So, the number is 2¹⁵*3¹⁰*5⁶ = 30,233,088,000,000


That's a pretty big number.

Well Drew you have escaped the fate worse than death !

It s tedious going thru your working, so Im giving you the method used for this problem, which, I would say is more precise, but much the same, so you may compare both yours and mine.*

As G.H. Hardy said that If the answer obtained by a method is correct then the method should be correct, whatever, unless you have the intuition of Ramanujan and have no method!

A number N divisible by 2, 3, and 5 may have the form N = 2^a*3^b*5^c.

Then since N/2 is a square 'a' must be odd and b and c even.

Similarly a and c must be multiples of 3 and b ==1 mod 3.

Also, a and b must be multiples of 5 and c==1 mod 5.

The samllest values satisfying these conditions are

a=15 , b= 10 , c= 6 and

N= 2^15*3^10*5^6 = 30,233,088,000,000.

* [Method by Albert H. Beiler]

As Wacky said that you have certainly 'zeroed in' on the number.

Excellent Drew! keep up the good work!

Mally