# 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

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

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 : -

This student written piece of work is one of many that can be found in our GCSE Pythagorean Triples section.

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