tudor

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a number that is between 100 and less then 200 and the number is 19 times larger than its sum

  • tudor -

    I assume you mean the sum of digits.

    let the number be xyz
    where x is 1
    (1+y+z)= 1yz/19

    Look at the right side. That has to be a whole number, so 1yz is a multiple of 19
    I can range from 6 to 9
    which means y+z can range from 5 to 8
    try 5=y+z or y=5-z which means if
    z; y;
    2;3
    3,2
    4,1
    1,4
    5,0
    0,5
    but 1yz=19*6 or 1yz/6=19, which means 1yz is divisible by six, so it ends in 6, 2, 8, 4
    if =2, y=4 142 /19=not six
    if z=4, y=2 124/19=not six
    so yz now runs from 7 to 8
    If 1+y+z= then combinations
    y+z=6 or 7
    assume y+z=6
    y, z
    0,6
    6,0
    1,5
    5,1
    4,2
    2,4
    Now try these combinations:
    1+0+6=7, 7*19=133 nope
    1+1+5=7, 133 does not match
    none of the y,z above will give 133

    Now try y+z=7
    6,1 1+6+1=8, 8*19=152 and I see immediately that y=5, z=2

    152 is the number
    sum digits: 8
    the number is 19 times the sum

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