Express the hcf of 234 and 111 as 234x and111y.where x and y are integers

Question

Express the hcf of 234 and 111 as 234x and111y.where x and y are integers

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Amelia 1 week 2021-09-10T21:44:10+00:00 2 Answers 0

Answers ( )

    0
    2021-09-10T21:45:50+00:00

    Answer:

    (- 9 × 234 ) + (19 × 111 )

    Step-by-step explanation:

    Using the division algorithm to find the hcf

    If a and b are any positive integers, then there exists unique positive integers q and r such that

    a = bq + r → 0 ≤ r ≤ b

    If r = 0 then b is a divisor of a

    Repeated use of the algorithm allows b to be found

    here a = 234 and b = 111

    234 = 2 × 111 + 12 → (1)

    111 = 9 × 12 + 3 → (2)

    12 = 4 × 3 + 0 ← r = 0

    Hence hcf = 3

    We can now express the hcf (d) as

    d = ax + by where x, y are integers

    From (2)

    3 = 1 × 111 – 9 × 12

    From (1)

    3 = 1 × 111 – 9( 1 × 234 – 2 × 111)

       = 1 × 111 – 9 × 234 + 18 × 111

       = – 9 × 234 + 19 × 111 ← in required form

    with x = – 9 and y = 19

    0
    2021-09-10T21:45:59+00:00

    Answer:

    The HCF = 3.

    Step-by-step explanation:

    The prime factors of

    234 = 2*3*3*13,

    and of 111 = 3*37.

    The only common factor is 3.

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