this 6 digit number = half the square of its last 3 figures . What is the number

Question

"Look,Dad" said Harry, pointing to the huge sign by the roadside, " What a funny number"

Greg smiled.  "Just the population of the city, an ordinary 6 digit number" he said " What's so odd about that."

"Something special" replied the boy.  "It's exactly half the square of its last three figures."

He was right. so what was that number?

Answer 1

Let the number be 1000A+B. So 1000A+B=B^2/2, where B<1000.

2000A+2B=B^2; B^2-2B+1=2000A+1; (B-1)^2=2000A+1. B=1+sqrt(2000A+1)<1000 and sqrt(2000A+1)<999. Therefore 2000A+1<998001 and 99<A<499. When A=125, sqrt(2000A+1)=500.001.

When A=125, B=501 and the number is 125501. B^2/2=125500.5 which rounded up to the nearest integer is 125501. So this is a close match.

The next close match is when A=180 and B=601, 1000A+B=180601 and B^2/2=180600.5.

Then we have: A=245, B=701; the number is 245701; 701^2/2=245700.5.

But when A=282, B=752, 282752=752^2/2. Exact match!

(Other near misses are: 320801 and 405901, )