CTET Paper II (Mathematics and Science) 15 Dec 2024 Shift 1 Solved Paper

Concept:Prime factorization reveals all prime factors and their powers in a product. A number is a factor only if it is made up of these primes with equal or lower powers.Explanation:Find the prime factorization of each number:$264 = 2^{3} \times 3 \times 11$$90 = 2 \times 3^{2} \times 5$$1680 = 2^{4} \times 3 \times 5 \times 7$Multiply them: $264 \times 90 \times 1680 = 2^{3+1+4} \times 3^{1+2+1} \times 5^{1+1} \times 7 \times 11 = 2^{8} \times 3^{4} \times 5^{2} \times 7 \times 11$Now test the smallest numbers from the options:12 = $2^{2} \times 3$ – both primes are present with enough power, so 12 is a factor.14 = $2 \times 7$ – both present, so 14 is a factor.15 = $3 \times 5$ – both present, so 15 is a factor.13 is a prime number not present in the factorization. Hence 13 cannot divide the product.Thus the smallest positive integer not a factor is 13.Answer:D. 13