Yahoo Answers is shutting down on May 4th, 2021 (Eastern Time) and beginning April 20th, 2021 (Eastern Time) the Yahoo Answers website will be in read-only mode. There will be no changes to other Yahoo properties or services, or your Yahoo account. You can find more information about the Yahoo Answers shutdown and how to download your data on this help page.

Is there a number like this?

Is there a 3-digit number that, when the three digits are multiplied together, equal that number?

Like, if the number was xyz, so x*y*z=xyz?

4 Answers

Relevance
  • 10 years ago

    The number xyz has a value of:

    x*10^2 + y*10 + z

    So the question we're asking is is there x,y,z , 0<=x,y,or z<=9 such that:

    x*y*z = x*10^2 + y*10 + z

    Dividing both sides by 10^2 gives:

    x*y*z / 100 = x + y/10 + z/100

    Note that since x, y, and z are all less than 10 and nonnegative, x*y*z / 100 <= x. By observing that

    x <= x + y/10 + z/100 we can conclude that:

    x*y*z / 100 <= x + y/10 + z/100

    We can strengthen <= to < by noting that if any one of x, y, or z are nonzero, through appropriate relabeling we are guaranteed to have x < x + y/10 + z/100. Therefore we get:

    x*y*z / 100 < x + y/10 + z/100

    From this inequality we can conclude that x*y*z cannot equal the number "xyz".

  • Alex
    Lv 6
    10 years ago

    xyz = 100x + 10y + z

    x*y*z = xyz

    ⇒ x*y*z = 100x + 10y + z

    If x = 1

    ⇒ the maximum value of x*y*z is when y = z = 9 so it's 81, but 100x + 10y + z = 100 + 10y + z, which is greater than 81

    ⇒ if the 1st digit is 1, there is no such number

    If x = 2

    ⇒ the maximum value of x*y*z is again when y = z = 9 so it's 162, but 100x + 10y + z = 200 + 10y + z, which is greater than 162

    ⇒ if the 2nd digit is 2, there is no such number

    For x = 3,4, ..., 9

    ⇒ similarly: x*y*z < 100x + 10y + z

    ⇒ there is no such a 3-digit number

  • 10 years ago

    Diophantine: a + 10b + 100c = a * b * c where a<10, b<10, c<10

    Only: 000

  • 10 years ago

    111.

Still have questions? Get your answers by asking now.