The digits and square numbers

Number Theory Level pending

All nine digits can be arranged to form four square numbers.

A , BC , DEF , GHI .

Find the sum of all these four numbers.

Use the hint if you need............... ............................,....

Hint : all the four numbers are divisible by 9 and their sum is also divisible by 9.

2 solutions

N K
Sep 21, 2020

With the hint it is obvious. But I began without hint and found 1+36+529+784=1350 (1^2+6^2+23^2+28^2). Why not?

Nashita Rahman
Sep 13, 2016

The numbers are 9 , 81, 324, 576. Each of the numbers are divisible by 9 and their sum is also divisible by 9.

So The answer is 990

How did you find these 4 numbers?

Pi Han Goh - 4 years, 9 months ago

Since all of are divisible by 9. So A=9.

The 2-digit square numbers divisible by 9 are 36, 81. So BC=36 or 81

Case 1: BC=36

The digits left are 1, 2, 4, 5, 7.

The 3-digit square numbers which are divisible by 9 using them are 144, 225.........

In this case the numbers obtained repeat the digits. So BC is not equal to 36.

Case 2: BC=81

The digits left are 2, 3, 4, 5, 6, 7

We see that 9×36 = 324 and 9×64 =576. Here all of them are square numbers (this was also hint which I didn't provide in the problem)

Hope I could answer!

Nashita Rahman - 4 years, 9 months ago

Actually the sum is obvious, if numbers are divisible by 9, their sum would also be!

Prince Loomba - 4 years, 5 months ago