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Subtract the following hexadecimal numbers: EBA16 - 2E16

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Answer

EBA16−2E16=E8C16EBA_{16} - 2E_{16} = E8C_{16}

BABA16−EBA16=AC0016BABA_{16} - EBA_{16} = AC00_{16}

Step-by-step solution

7 steps

Okay CØTBØJ, let's tackle these hexadecimal subtraction problems.

Hexadecimal (base 16) uses digits 0-9 and letters A-F, where:
A = 10
B = 11
C = 12
D = 13
E = 14
F = 15

When you need to borrow in hexadecimal subtraction, you borrow 1 from the next column to the left, which adds 16 to the current column's value.


Problem (c):

EBA16−2E16EBA_{16} - 2E_{16}

We'll subtract column by column, from right to left.

  1. Subtract the rightmost column (16⁰ place).

    We need to calculate A16−E16A_{16} - E_{16}.
    In decimal, this is 10−1410 - 14. Since 10<1410 < 14, we need to borrow from the next column.
    Borrow 1 from B in the 16116^1 place. B (111011_{10}) becomes A (101010_{10}).
    The borrowed 1 is equivalent to 161016_{10} in the current column. So, A (101010_{10}) becomes 10+16=261010 + 16 = 26_{10}.
    Now, subtract: 2610−E16=2610−1410=121026_{10} - E_{16} = 26_{10} - 14_{10} = 12_{10}.
    In hexadecimal, 121012_{10} is C.
    So, the rightmost digit is C.

  2. Subtract the middle column (16¹ place).

    The B in EBA16EBA_{16} became A after borrowing.
    We need to calculate A16−216A_{16} - 2_{16}.
    In decimal, this is 10−2=81010 - 2 = 8_{10}.
    In hexadecimal, 8108_{10} is 8.
    So, the middle digit is 8.

  3. Subtract the leftmost column (16² place).

    We need to calculate E16−016E_{16} - 0_{16} (since there's no digit in the 16216^2 place for 2E162E_{16}).
    In decimal, this is 14−0=141014 - 0 = 14_{10}.
    In hexadecimal, 141014_{10} is E.
    So, the leftmost digit is E.

    Combining the results:

    EBA16−2E16=E8C16EBA_{16} - 2E_{16} = \boxed{E8C_{16}}

    Problem (d):

    BABA16−EBA16BABA_{16} - EBA_{16}

    We'll subtract column by column, from right to left.

  4. Subtract the rightmost column (16⁰ place).

    We need to calculate A16−A16A_{16} - A_{16}.
    In decimal, this is 10−10=01010 - 10 = 0_{10}.
    So, the rightmost digit is 0.

  5. Subtract the second column from the right (16¹ place).

    We need to calculate B16−B16B_{16} - B_{16}.
    In decimal, this is 11−11=01011 - 11 = 0_{10}.
    So, the second digit from the right is 0.

  6. Subtract the third column from the right (16² place).

    We need to calculate A16−E16A_{16} - E_{16}.
    In decimal, this is 10−1410 - 14. Since 10<1410 < 14, we need to borrow from the next column.
    Borrow 1 from B in the 16316^3 place. B (111011_{10}) becomes A (101010_{10}).
    The borrowed 1 is equivalent to 161016_{10} in the current column. So, A (101010_{10}) becomes 10+16=261010 + 16 = 26_{10}.
    Now, subtract: 2610−E16=2610−1410=121026_{10} - E_{16} = 26_{10} - 14_{10} = 12_{10}.
    In hexadecimal, 121012_{10} is C.
    So, the third digit from the right is C.

  7. Subtract the leftmost column (16³ place).

    The B in BABA16BABA_{16} became A after borrowing.
    We need to calculate A16−016A_{16} - 0_{16} (since there's no digit in the 16316^3 place for EBA16EBA_{16}).
    In decimal, this is 10−0=101010 - 0 = 10_{10}.
    In hexadecimal, 101010_{10} is A.
    So, the leftmost digit is A.

    Combining the results:

    BABA16−EBA16=AC0016BABA_{16} - EBA_{16} = \boxed{AC00_{16}}
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