6th grader Math Olympiad quiz for the 'Super Genius'

[YiJiun]

Hi, actually I am not too sure if this question belongs to the Arithmetic section, but since this question was designed for 6th graders, I'm making just a guess here. We derived at the answer through using the approach of "trial and error" which is unreasonably tedious and impractical.

The question: In the following equation, each letter represents a distinct digit. 7(ABCXYZ) = 6(XYZABC). A and X are not equal to zero. Find the six digit number ABCXYZ.

Please explain a simpler and easier method to the solution (suitable for 6th graders). Thank you.



Our thought process: 7(ABCXYZ) = 6(XYZABC) = multiple of 42.
ABCXYZ = multiple of 6; XYZABC = multiple of 7

 

[Deleted member 4993]

Hi, actually I am not too sure if this question belongs to the Arithmetic section, but since this question was designed for 6th graders, I'm making just a guess here. We derived at the answer through using the approach of "trial and error" which is unreasonably tedious and impractical.

The question: In the following equation, each letter represents a distinct digit. 7(ABCXYZ) = 6(XYZABC). A and X are not equal to zero. Find the six digit number ABCXYZ.

Please explain a simpler and easier method to the solution (suitable for 6th graders). Thank you.



Our thought process: 7(ABCXYZ) = 6(XYZABC) = multiple of 42.
ABCXYZ = multiple of 6; XYZABC = multiple of 7

Quick thoughts:

let M = ABC & N = XYZ then

7(1000M + N) = 6(1000N + M) → 6994M = 5993N

6994 = 13*538

5993 = 13* 461

Don't know where I am going with all these - need to think more.....

 

[Deleted member 4993]

Quick thoughts:

let M = ABC & N = XYZ then

7(1000M + N) = 6(1000N + M) → 6994M = 5993N

6994 = 13*538

5993 = 13* 461

Don't know where I am going with all these - need to think more.....

I see that

M = 461 and N = 538 will satisfy the conditions.

6*(538461) = 7 * (461538)

Hence the number is 461538 (it is the unique solution because 461 & 538 are relatively prime and 2*538 is not a 3 digit number)