Integer solutions for equations A * [(B + C)(D - E) - F(G*H) ] / J = 1

[Ben Fido]

Could anyone please answer this


You are confronted with the following formula:

A * [(B + C)(D - E) - F(G*H) ] / J = 10

Knowing that each variable is a unique, single-digit, nonzero number, and that C - B = 1, and H - G = 3, what is the number ABCDEFGHJ, where each letter is a digit? For example, if A = 4, B = 2, and C = 7, ABC would equal 427.

Thanks

 

[arildno]

You have 3 equations and 9 unknowns, so most likely, you have lots of solutions.

Let us set A=J, H=3, G=0, whereby we must have (B+C)(D-E)=10

Setting now B=2, C must be 3, and D-E=2.

These particular choices determine 720 distinct solutions to your problem.

Or should your numbers be non-repeating? (Your uniqueness condition?)

Okay, I forgot that none of your numbers should be zero, either.

Then your problem is difficult, in the sense of being tedious:
If I have counted correctly, there exist 26 different quadruples (B,C,G,H) so that C-B=1, H-G=3, and all 4 numbers are unique, single digit numbers.

You will solve your problem on a trial&error basis restricting yourself to these 26 special cases.

 

[Ben Fido]

Aw come on, I just want the answer in a hurry!

It is Christmas, afer all.

 

[arildno]

Well, since there will exist for each such quadruple 10 combinations of D and E (since D-E must be positive), and the three remaining numbers can be placed in the A, J F positions, you are free to analyze the 26*10*6=1560 special cases occurring.

But don't count upon us to do it for you.

 

[malawi_glenn]

Ben Fido, please read and follow the rules for these forums.