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If \(f_{1} = 6x^{2}y\) and \(f_{2} = 15xy^{2}\), then which of the following option is true ?

  • A.H.C.F. \((f_{1}, f_{2}) = 45xy\), L.C.M. \((f_{1}, f_{2}) = 30x^{2}y^{2}\)
  • B.H.C.F. \((f_{1}, f_{2}) = 30xy\), L.C.M. \((f_{1}, f_{2}) = 30x^{3}y^{3}\)
  • C.H.C.F. \((f_{1}, f_{2}) = 3xy\), L.C.M. \((f_{1}, f_{2}) = 90x^{2}y^{2}\)
  • D.H.C.F. \((f_{1}, f_{2}) = 3xy\), L.C.M. \((f_{1}, f_{2}) = 30x^{2}y^{2}\)

Answer: D

The true option is H.C.F. = 3xy, L.C.M. = \(30x^{2}y^{2}\).

Write \(f_{1} = 6x^{2}y = 2 \times 3 \times x^{2}y\) and \(f_{2} = 15xy^{2} = 3 \times 5 \times xy^{2}\).

The H.C.F. takes the common factors with the lowest powers: \(3 \times x \times y = 3xy\). The L.C.M. takes every factor with the highest power: \(2 \times 3 \times 5 \times x^{2}y^{2} = 30x^{2}y^{2}\). Check: \(3xy \times 30x^{2}y^{2} = 90x^{3}y^{3} = f_{1}f_{2}\).

45xy and 30xy cannot be common factors of 6 and 15, \(30x^{3}y^{3}\) uses too high powers, and \(90x^{2}y^{2}\) multiplies the numbers instead of taking their L.C.M.

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