I solved both the questions without even thinking of the word prime factor. My thought process was like this:

Q1: Take x as 6a and y as 14b. The product is 84ab. Having x as a multiple of 9 would mean xy would be 84ab * 9, which cannot be a multiple of 105 since it doesnt end with 5 or 0. If is a multiple of 25, that means xy is 84*25ab, which is a multiple of 7 and 15 (7*15 = 105). Hence b is sufficient.

For the second question, my thought process veered towards using the word product. Given that red has 7 points, I just started dividing 147000 by 7 and got 21000, which I again divided by 7 and got 3000, which cannot be further divided by 7. Hence, the answer is 2 red beads.

Please note that even though I did not use prime factors explicitly, my knowledge of multiplication and factors in general helped me answer this. By using prime factors, you are just making it more systematic.

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