a box contains more than 25 but less than 40 balls.when we counted by 3 there are 1 left over,when we counted by 5 there are 2 left over.how many balls are there in the box ? please explain how you work it out,thanks

there aren't that many multiples of 5 between 25 and 40: 3 and ,35

it's easy to check that 37 = 7*5+2 = 12*3+1

To solve this problem, we can use the method of finding the least common multiple (LCM) of the two numbers 3 and 5. The LCM is the smallest multiple that is divisible by both numbers.

First, let's find the LCM of 3 and 5:

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, ...

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, ...

From the lists above, we can see that the LCM of 3 and 5 is 15. So, any number that leaves a remainder of 1 when divided by 3 and a remainder of 2 when divided by 5, will also leave a remainder of 1 when divided by 15.

Now, let's find the number of balls that satisfies these conditions.

Starting from 26 (to make sure it's greater than 25) and going up by 15, we can find the following numbers that satisfy the conditions:

26, 41, 56, 71, ...

But we are looking for a number that is less than 40. So, the only possible solution is 26.

Therefore, there are 26 balls in the box.

Explanation:
- To solve this problem, we used the method of finding the least common multiple (LCM) to determine the pattern of counting the balls.
- By finding the LCM of 3 and 5, we discovered that any number that satisfies the conditions for both remainsders will also satisfy the condition for the LCM.
- We then checked the numbers starting from 26 (greater than 25) and going up by 15 (the LCM), until we found a number that is less than 40.
- The only number that satisfies the conditions and is less than 40 is 26, so that is the solution.