Math Is Fun Forum

Lacy21580

Member

Registered: 2010-10-23

Posts: 1

How do we solve ABCDEF × 3 = BCDEFA

OK, I know it's been a while for me using my brain creatively, but can you please explain how you came up with the answers that are shown below for the given problem. I have know idea how to begin to figure this out.

The Puzzle: Solve the following:             

ABCDEF × 3 = BCDEFA

The Solution . . .

285714 × 3=857142
or
142857 × 3=428571

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: How do we solve ABCDEF × 3 = BCDEFA

Hi Lacy21580;

You do these type problems usually by computer when you are unlucky enough to have to do them by hand you have to prune the possibilites down by using arthmetic. For instance A has to be less than 4 because otherwise 3 * a six digit number would be a 7 digit number. A should not be 3 otherwise B would definitely be a 9 and 3 times a 9 would cause a 2 carry into the A which would mean A would be bigger than 2 which is illegal. So we have

A ={ 1 , 2 }

If A = 1 then F = 7 and If A = 2 then F = 4

So we have the 2 cases

1 B C D E 7 * 3 = B C D E 7 1

or

2 B C D E 4 * 3 = B C D E 4 2

In the first the 3 * 7 produces a 2 carry 3 E + 2 = 7 so E = 5

In the second 3 * 4 produces a 1 carry 3 E + 1 = 4 so E = 1

So we now have:

1 B C D 5 7 * 3 = B C D 5 7 1

or

2 B C D 1 4 * 3 = B C D 1 4 2

Now for the first one we have 3 * D = 5 and the carry from last column makes it 3*D + 1 = 5 so D = 8

For the second one we have 3 * D = 1 and no carry from last column makes it 3*D  = 1 so D = 7

So we now have:

1 B C 8 5 7 * 3 = B C 8 5 7 1

or

2 B C 7 1 4 * 3 = B C 7 1 4 2

Now for the first one we have 3 * C = 8 and the carry from last column makes it 3*C + 2 = 8 so C = 2

For the second one we have 3 * C = 7 and no carry from last column makes it 3*C + 2 = 7 so C = 5

So we now have:

1 B 2 8 5 7 * 3 = B 2 8 5 7 1

or

2 B 5 7 1 4 * 3 = B 5 7 1 4 2

Now you have 2 separate linear equations to solve.

B = 4

And

B = 8 So,

1 4 2 8 5 7 * 3 = 4 2 8 5 7 1

or

2 8 5 7 1 4 * 3 = 8 5 7 1 4 2

It is much easier by computer!

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: How do we solve ABCDEF × 3 = BCDEFA

Hi Lacy21580 and bobbym

Cannot improve on that but here's an interesting spinoff.

1/7 = 0.142857142857........
2/7 = 0.285714285714........
3/7 = 0.428571428571........
4/7 = 0.571428571428........
5/7 = 0.714285714285........
6/7 = 0.857142857142........

So if ABCDEF = 142857  then BCDEFA = 3 x ABCDEF as 3 x 1/7 = 3/7 etc.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob