If A = x³y² and B = xy³, then find the HCF of A and B.
2025
If A = x³y² and B = xy³, then find the HCF of A and B.
- A.
x²y²
- B.
xy³
- C.
xy²
- D.
none
Show answer & explanation
Correct answer: C
The HCF (highest common factor) of two monomials that involve only shared variables is obtained by taking, for each variable common to both expressions, the smaller of the two exponents in which it appears, and multiplying these together. A variable that is missing from either expression cannot appear in the HCF at all.
In A = x3 y2, the exponent of x is 3 and the exponent of y is 2. In B = x y3, the exponent of x is 1 (since x is the same as x1) and the exponent of y is 3.
Compare the two exponents of x: 3 in A and 1 in B — the smaller of the two is 1.
Compare the two exponents of y: 2 in A and 3 in B — the smaller of the two is 2.
Combine these smaller exponents: x raised to the power 1, times y raised to the power 2.
Confirm by division: A divided by x y2 leaves x2, a monomial with no negative or fractional powers, and B divided by x y2 leaves y, also a clean monomial — so x y2 divides both A and B exactly. Raising either exponent by one more power would fail this division for one of the two expressions, confirming x y2 is the highest common factor.
