Discussion
What is the smallest number by which 3,600 must be divided to make it a perfect cube?
Answer: B
Correct answer: 450
Factorise into primes: \[ 3600 = 2^{4} \times 3^{2} \times 5^{2}. \] A perfect cube requires every exponent to be a multiple of 3. Here the exponents are 4, 2 and 2. Dividing must reduce each to the nearest lower multiple of three - so \( 2^{4} \) becomes \( 2^{3} \), and \( 3^{2} \) and \( 5^{2} \) must go entirely, since their next lower multiple is \( 3^{0} \) and \( 5^{0} \). The divisor is therefore \[ 2^{1} \times 3^{2} \times 5^{2} = 2 \times 9 \times 25 = 450. \] Checking: \( 3600 \div 450 = 8 = 2^{3} \), a perfect cube. The companion question - the smallest number to multiply by - raises each exponent to the next multiple of three instead, giving \( 2^{2} \times 3^{1} \times 5^{1} = 60 \), and \( 3600 \times 60 = 216000 = 60^{3} \). The same method with multiples of two answers the equivalent questions about perfect squares.
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