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Math Reflections

Posted by JB on April 5, 2012 – 7:55 pm

What 5 digit number when multiplied by 54321 produces a 10 digit number end with 12345?

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This post is under “Tom” and has 4 respond so far.
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4 Responds so far- Add one»

  1. 1. Wizard of Oz Said:

    99945

  2. 2. Krazeedude Said:

    99945

    *was beaten by miles*

  3. 3. Knightmare Said:

    hi Krazeedude…you got to tell Miles not to treat you like that, no matter how much he wants to know the answer

  4. 4. Chris Said:

    I can’t find a nice (e.g. modulo arithmetic technique). So here’s a direct method.

    Let the number be abcde i.e. abcde*54321 = y*10^5 + 12345
    We don’t care what y is, so I’ll completely ignore it.

    You might find it useful to practice long multiplications to follow this solution.
    The method is to determine e, then use that to determine d, etc.

    Do a long multiplication

    abcde
    54321
    -----
    abcde i.e. 1*abcde
    ????0 i.e. 20*abcde
    ???00 i.e. 300*abcde
    ??000 i.e. 4000*abcde
    ?0000 i.e. 50000*abcde
    -----
    12345
    
    So e+0+0+0+0 = 5 => e = 5
    So use that and re-examine
    
    abcd5
    54321
    -----
    abcd5 1*abcd5
    ???00 20*5 = 100
    ???00
    ??000
    ?0000
    -----
    12345
    
    So d+0+0+0+0 = 4 => d = 4.
    
    Substitute and retry.
    
    abc45
    54321
    -----
    abc45 1*45 = 45
    ??900 20*45 = 900
    ??500 300*45 = 13500
    ??000
    ?0000
    -----
    12345
    
    We need c + 9 + 5 = 3 (mod 10)
    => c = -11 = 9 (mod 10)
    So substituting for c =>
    
    ab945
    54321
    -----
    ab945 1*945 = 945
    ?8900 20*945 = 18900
    ?3500 300*945 = 283500
    ?8000 4000*945 = 3780000
    ?0000
    -----
    12345
    
    For the c column we have 9+9+5 = 23, so we carry 2 to the b column.
    
    So we need 2 = 2 + b + 8 + 3 (mod 10) => b = 9
    
    Rinse and repeat
    
    a9945
    54321
    -----
    a9945
    98900 20*9945 = 198900. 20*a0000 contributes nothing
    83500 300*9945 = 2983500
    80000 4000*9945 = 39780000
    50000 50000*9945 = 497250000
    -----
    12345
    
    For the b column we have 9+8+3 = 20, so we carry 2 to the a column.
    
    Need 1 = 2 + a + 9 + 8 + 8 + 5 (mod 10) => a = 9
    
    Hence abcde = 99945
    

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