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Math Help - Linear Diophantine Equations

  1. #1
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    Linear Diophantine Equations

    I am having difficulties with this problem:

    Is it possible to have 50 coins, all of which are pennies, dimes, or quarters, with a total worth $3?

    Thank you!
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  2. #2
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    Quote Originally Posted by jmc1979 View Post
    I am having difficulties with this problem:

    Is it possible to have 50 coins, all of which are pennies, dimes, or quarters, with a total worth $3?

    Thank you!
    Let,
    x # of pennies.
    y # of dimes.
    z # of quaters.

    Henceforth,
    x+y+z=50
    x+10y+25z=300

    Thus,
    (50-y-z)+10y+25z=300
    9y+24z=250
    Note that,
    gcd(9,24)=3 and 3 does not divide 250.
    Thus, there can be no such combination of coins.
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  3. #3
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    Hello, jmc1979!

    Is it possible to have 50 coins, all of which are pennies, dimes, or quarters, with a total worth $3?

    Let P = number of pennies.
    Let D = number of dimes.
    Let Q = number of quarters.

    A total of 50 coins: . [1] . . P + D + Q . . = . 50

    Total value of 300: .[2] .P + 10D + 25Q .= .300


    From [1], we have: .Q .= .50 - P - D

    Substitute into [2]: .P + 10D + 25(50 - P - D) .= .300

    This simplifies to: .24P + 15D .= .950 . . 3(8P + 5D) .= .950

    The left side is a multiple of 3 . . . The right side is not.
    . . An impossible situation . . .

    The answer to the question is: .No


    Curses . . . too slow again!
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