Hi, i'm preparing for Olevels and i have some maths questions that i'm not sure how to do, i'd appreciate it if you could help.

1. The numbers 60 and 126, written as products of their prime factors:
60=2^2x3x5
126=2x3^2x7

a)Find the largest integer which is a factor of both 60 and 126.
b)Find the smallest integer which is an exact multiple of both 60 and 126.

How do i make use of the prime factors that the question provided?? I really don't get it...

2. What does 'upper bound' mean?
This came from a maths question that says something like "The length of a rectangle is 12cm, correct to the nearest centimeter. What is the upper bound for the length of the rectangle?". Does it mean like, what is the largest possible length of the rectangle before it was rounded off?

3. Three bus services operate from the same depot. The first service leaves at 10 minute intervals, the second at 15 minute intervals and the third at 25 minute intervals. All three buses leaves at the 8am, at what time does does the three buses depot together?
*** I think i know how to find the answer, but i'm not sure if there's a faster way of doing. I only know how to list down all the multiples of the numbers and check for the first one common multiply. But that would be a rather slow method isn't it??

P.S. so basically i have problems with lowest common multiple and highest common factor. Please help! Thanks!!!!

Hi! I'd be happy to help you with your math questions.

1. For the first question, you are given the prime factorization of both numbers: 60 and 126. To find the largest integer that is a factor of both numbers, you need to find the common factors they have in their prime factorizations.

To do this, you can compare the prime factors of the two numbers. The common factors will be the ones that appear in both factorizations. In this case, the common factors are 2 and 3.

To find the largest integer factor, you can take the product of these common factors. In this case, it would be 2 x 3 = 6.

For the second part of the question, to find the smallest integer multiple, you need to find the smallest multiple that both numbers have in common. In this case, you need to find the least common multiple (LCM) of 60 and 126.

To find the LCM, you can multiply the highest powers of all the prime factors involved. So, in this case, it would be 2^2 x 3^2 x 5 x 7 = 2 x 2 x 3 x 3 x 5 x 7 = 2520.

Therefore, the largest integer that is a factor of both 60 and 126 is 6, and the smallest integer that is an exact multiple of both 60 and 126 is 2520.

2. In the context of your question, an "upper bound" refers to the maximum value that a particular quantity can have. When you are asked to find the upper bound of the length of a rectangle, it means finding the largest possible length it could have before it was rounded off.

For example, if the length is given as 12cm, correct to the nearest centimeter, the upper bound would be the largest length that is still within 1cm of 12cm. In this case, it would be 12cm + 0.5cm = 12.5cm.

So, the upper bound for the length of the rectangle in this case would be 12.5cm.

3. To find the time when the three buses from the same depot leave together, you need to find the least common multiple (LCM) of their intervals.

You mentioned that the first bus leaves every 10 minutes, the second bus every 15 minutes, and the third bus every 25 minutes. To find the LCM, you can follow a similar process as in the first question.

You can start by listing down the multiples of each interval until you find the first common multiple. However, there is a more efficient way to find the LCM.

You can find the LCM by considering the prime factors of the intervals. In this case, the prime factorization of 10 is 2 x 5, 15 is 3 x 5, and 25 is 5 x 5. To find the LCM, you can take the highest power of each prime factor involved. In this case, it would be 2 x 3 x 5 x 5 = 150.

Therefore, the three buses from the same depot will leave together at 8:00 am + 150 minutes, which is at 10:30 am.

I hope this explanation helps! Let me know if you have any further questions.