Final answer:
The smallest positive integer 'n' that when multiplied by 140 results in a squared integer is 35. To make 140 a perfect square, its unpaired prime factors, 5 and 7, need to be paired, and multiplying by 35 accomplishes this.
Step-by-step explanation:
The question asks to find the smallest positive integer 'n' such that the product of 140 and 'n' is the square of an integer. To determine this, we need to factorize 140 and see which smallest positive integer when multiplied by 140 will make it a perfect square.
Firstly, the prime factorization of 140 is 2² × 5 × 7. For it to be a perfect square, all prime factors must occur in pairs. Currently, the factors 5 and 7 are not paired. Therefore, the smallest integer 'n' that will create the necessary pairs for 5 and 7 is 5 × 7 or 35.
When we multiply 140 by 35, we get: 140 × 35 = 2² × 5² × 7², which is indeed the square of the integer 70 (70²).
The answer is thus (d) 35.