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3 Digit Number - Maths Investigations

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Introduction

CWK1

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=2223330 ÷ 15=2225328 ÷ 24=222

  147                        258                        369

  174                        285                        396

  471                        528                        639

  417                        582                        693

  741                        825                        936

+714 _                          +852                          +963

2664÷ 12=2223330 ÷ 15=2223996 ÷ 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=1111110 ÷ 10=1111887 ÷ 17=111

  224                        559                        772

  242                        595                        727

+422                          +955                          +277

888÷ 8=1112109 ÷19=111        1776 ÷ 16=111

...read more.

Middle

=111

What if all 3 digits are the same number?

If All Digits Are The Same : -

333    = 37444    = 37666        = 37999        = 37777 _        = 37

  9                 12                 18                   27                 21

It seems that when all the 3 digits are the same number the answer to the problem is 37.

Can I use Algebra to explain this?

aaa=100a+10a+a

=111a             = 37

 3a

What if

...read more.

Conclusion

+cbaa=1000c+100b+10a+a

6666a+3333b+3333c

2a+b+c

=3333(2a+b+c) =3333

2a+b+c

What if 3 of the 4 digits are the same?

If 3 Of The 4 Digits Are The Same : -

 4447

 4474

 4744

+7444

21109÷19=1111

Is this the same answer every time 3 of the 4 digits are the same?

Lets usr Algebraic terms to find out.

aaab=1000a+100a+10a+b

aaba=1000a+100a+10b+a

abaa=1000a+100b+10a+a

baaa=1000b+100a+10a+a

3333a+1111b

3a+b

=1111(3a+b)        =1111

3a+b

What if all 4 digits are the same?

If All 4 Digits Are The Same : -

9999        =277.75

          36

Is this the same answer every time all 4 digits are the same?

Let’s use Algebraic terms to find out.

aaaa=1000a+100a+10a+a

=1111a        =277.75

        4a

What if I do the same problem but with 5 digits instead of 4 or 3?

From the data I have gathered maybe I can guess the outcomes for 5 digits.

5 Digit Number

If All 5 Digits Are Different : -

...read more.

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