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ABC + DEF + GHI = JJJ, what does J represents?

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Consider the sum:
ABC + DEF + GHI = JJJ
If different letters represent different digits, and there are no leading zeros,

What does J represent?

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SherlockHolmes Expert Asked on 29th September 2015 in Math Puzzles.
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    • 1 Answer(s)
    The value of J is 9

     

    EXPLANATION-

    As there are no leading zeros, J must be 7, 8, or 9. (JJJ = ABC + DEF + GHI = 14? + 25? + 36? = 7??)

    Now, the remainder left after dividing any number by 9 is the same as the remainder left after dividing the sum of the digits of that number by 9. Also, note that 0 + 1 + … + 9 has a remainder of 0 after dividing by 9 and JJJ has a remainder of 0, 3, or 6.

     
    The number 9 is the only number from 7, 8 and 9 that leaves a remainder of 0, 3 or 6 if you remove it from the sum 0 + 1 + … + 9.
    Hence, it follows that J must be 9.
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    Detective Expert Answered on 29th September 2015.
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