What is the number?

Question

The sum of the squares of the digits in a 3 digit number is 98.The sum of the first digit and the last digit is the middle digit. If the digits were reversed,the number is increased by 198.What is the number?

Answer 1

Let the number be 100a+10b+c. We know a+c=b and a²+b²+c²=98 and 100c+10b+a-100a-10b-c=99(c-a)=198, meaning c-a=2. So we can write c=a+2, therefore a+a+2=b and b=2a+2.

Sum of squares: a²+4a²+8a+4+a²+4a+4=98 when we substitute for b and c.

Therefore 6a²+12a-90=0=a²+2a-15=(a-3)(a+5), and a=3. So b=2a+2=8 and c=a+2=5.

The number is 385.

CHECK

3²+8²+5²=9+64+25=98, 583-385=198.