A "Perfect" Number (496 follower problem)

Number Theory Level 4

The number 496 is the third smallest perfect number: the sum of its proper divisors is the number itself: $1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496.$

But it is also a "perfect" number in the sense that its third digit is the geometric mean of the first two digits: $\sqrt{4\cdot 9} = 6$ .

How many three-digit integers $\overline{abc}$ are there with the property, that $c$ is the geometric mean of $a$ and $b$ ?

Note : Numbers starting in zero do not count. Thus, a "three-digit integer" lies between $100$ and $999$ .

And here is a more challenging variation on the theme.

The answer is 26.

2 solutions

Arjen Vreugdenhil
Jan 21, 2018

It is sufficient and necessary that $a\cdot b$ is a perfect square. There are two sets of nine obvious solutions: $\overline{abc} = 100, 200, \cdots, 900\ \ \ \ \text{and}\ \ \ \ 111, 222, \cdots, 999.$ Apart from these, there are four pairs of solutions (exchanging $a$ and $b$ doesn't change anything): $\overline{abc} = 142, 412;\ 193,913;\ 284,824;\ 496,946.$ Thus we find a total of $9 + 9 + 8 = \boxed{26}$ solutions.

Oh, I missed out the 222, 333 solutions. I had it as "a and b need to be perfect squares".

Calvin Lin Staff - 3 years, 4 months ago

Then you also missed the 284 en 824?

Arjen Vreugdenhil - 3 years, 4 months ago

Haha. Yup.

Calvin Lin Staff - 3 years, 4 months ago

@Calvin Lin, I missed the 100,200 solutions and got 17 solutions. But I didn't miss the 284 and 824 one.

Alex Fullbuster - 2 years, 1 month ago

Giorgos K.
Feb 5, 2018

Mathematica

Length@Select[Range[100, 999],GeometricMean[(s=IntegerDigits[#])[[{1,2}]]]==s[[3]]&]

Here is some code in C that does the job:

1 2	`#include<stdio.h> int main() { for (int c=0, n=100; n<1000; n++) if ((n%10)(n%10)==(n/10)%10(n/100)) printf("%d: %d\n",++c,n); }`

This code loops only through 65 values instead of 900:

1
2
3

#include<stdio.h>
int main() { int n=0; for (int c=0,cc=0; c<=9; c++,cc=c*c)
    for (int a=(cc+8)/9; a<=9; a++) if (a && cc%a==0) printf("%d: %d%d%d\n",++n,a,cc/a,c); }

Arjen Vreugdenhil - 3 years, 4 months ago

A "Perfect" Number (496 follower problem)

The answer is 26.

2 solutions

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