1. **State the problem:** (i) Express 1936 as the product of its prime factors using prime factorisation. (ii) Explain why 1936 is a perfect square using the prime factorisation. (iii) Given that both $m$ and $n$ are prime numbers, find $m$ and $n$ such that $\frac{1936 \times n}{m}$ is a perfect cube. 2. **Prime factorisation formula and rules:** Prime factorisation means expressing a number as a product of prime numbers. A perfect square has all prime factors with even powers. A perfect cube has all prime factors with powers multiples of 3. 3. **Prime factorisation of 1936:** Divide 1936 by 2 repeatedly: $$1936 \div 2 = 968$$ $$968 \div 2 = 484$$ $$484 \div 2 = 242$$ $$242 \div 2 = 121$$ Now 121 is not divisible by 2, try next prime 11: $$121 \div 11 = 11$$ $$11 \div 11 = 1$$ So prime factors are: $$1936 = 2 \times 2 \times 2 \times 2 \times 11 \times 11 = 2^{4} \times 11^{2}$$ 4. **Explain why 1936 is a perfect square:** Since the powers of prime factors are $4$ and $2$, both even numbers, 1936 is a perfect square. 5. **Find $m$ and $n$ such that $\frac{1936 \times n}{m}$ is a perfect cube:** We want: $$\frac{1936 \times n}{m} = \text{perfect cube}$$ Substitute prime factors: $$\frac{2^{4} \times 11^{2} \times n}{m}$$ Since $m$ and $n$ are prime, let $m = p$ and $n = q$ where $p,q$ are primes. For the expression to be a perfect cube, the powers of 2 and 11 after division and multiplication must be multiples of 3. Check prime factors 2 and 11 powers: - Current powers: 2 has power 4, 11 has power 2. - After multiplying by $n$ (which is prime), the power of that prime increases by 1. - After dividing by $m$ (which is prime), the power of that prime decreases by 1. Try $m=2$ and $n=11$: $$\frac{2^{4} \times 11^{2} \times 11}{2} = 2^{4-1} \times 11^{2+1} = 2^{3} \times 11^{3}$$ Both powers are 3, which is a multiple of 3, so this is a perfect cube. **Final answers:** (i) $1936 = 2^{4} \times 11^{2}$ (ii) 1936 is a perfect square because all prime powers are even. (iii) $m=2$, $n=11$ make $\frac{1936 \times n}{m}$ a perfect cube.