If ABC+DEF+GHI=JJJ, each letter stands for a different digit, and no number starts with zero, what is J?
(In reply to Answer
by K Sengupta)
-> ABC+DEF+GHI+10J =JJJ + 10J = 111J+10J = 121J
-> ABC+DEF+GHI+J =4J(mod (9).....(#)
Now, the letters A to J (in order) taken together corresponds to a permutation of the digits 0 to 9 (in order), and accordingly:
ABC+DEF+GHI+J = A+b+C+D+E+F+G+H+I+J(mod 9)
= 0+1+2+3+4=5+6+7+8+9 (mod 9)
= 45 (mod 9)
= 0 (mod 9)
Accordingly, from (#), we must have:
4J = 0(mod 9)
Since gcd(4,9) = 1, it folows that:
J = 0(mod 9)
Thus J is a decimal digit which is divisible by 9. Accordingly, J = 0, or 9. Since, none of the numbers can contain any leading zero, we observe that J=0 leads to a contradiction. Therefore, J=9.
It now remains to verify that there is at least one valid solution to the given problem. By means of trial and error, we observe that indeed there are many valid solutions to the given problem. four of the said solutions are as follows:
Consequently, the required value of J is 9.