# 3 Digit Number - Maths Investigations

## 3 Digit Number

Take any 3 digit number, write down all possible numbers that can be made with the three digits, add them up, divide the total by the sum of the 3 digits. Investigate.

If All Digits Are Different : -

123                        456                        789

132                        465                        798

213                        546                        879

231                        564                        897

321                        654                        987

+312                          +645                          +978

1332 ÷ 24=222        3330 ÷ 15=222        5328 ÷ 24=222

147                        258                        369

174                        285                        396

471                        528                        639

417                        582                        693

741                        825                        936

+714 _                          +852                          +963

2664 ÷ 12=222        3330 ÷ 15=222        3996 ÷ 18=222

It seems that when all 3 digits are different the answer to the problem is 222.

Can I use Algebra to explain this?

abc=100a+10b+c

acb=100a+10c+b

bac=100b+10a+c

bca=100b+10c+a

cba=100c+10b+a

+cab=100c+10a+b

222a+222b+222c = 222(a+b+c)

a+b+c

=222

What if 2 of the 3 digits are the same?

If 2 digits are the same : -

223                        334                        566

322                        343                        656

+232                          +433                          +665

777 ÷ 7=111                1110 ÷ 10=111        1887 ÷ 17=111

224                        559                        772

242                        595                        727

+422                          +955                          +277

888 ÷ 8=111                2109 ÷19=111                1776 ÷ 16=111