The storekeeper found that if he removed the apples from his box either 2,3,4,5 or 6 at a time there was always one left over. However, if he removed the apples 7 at a time there was no apples left. If the box holds 500 apples, how many apples does the storekeeper have?

Question

The storekeeper found that if he removed the apples from his box either 2,3,4,5 or 6 at a time there was always one left over. However, if he removed the apples 7 at a time there was no apples left. If the box holds 500 apples, how many apples does the storekeeper have?

Answer 1

If you are working on number theory, look up Chinese remainder theorem.

If you have a puzzle or problem to solve, you can do it as follows:

If we took one apple from the box, then the number of apples must be divisible by 2, 3, 4, 5, and 6. Therefore the number of apples would be 60 or its multiples. (LCM of 2,3,4,5,6)

We look for a number (60 or its multiples) so that if we divide this number by 7, we should have 6 left, since we took one apple out earlier.

Try
60/7=8 with remainder 4 (or 8R4).
120/7=17R1, 180=25R5, 240=34R2, 300=42R6 (Good).

If we now throw back our apple to make 301, then it would satisfy all the given conditions.

Answer 2

Since the LCM of 2,3,4,5,6 is 60, there are 60n+1 apples in the box.

That number must also be a multiple of 7, so since 60 = 7*8+4, 5*60+1 = 5*7*8+5*4+1 = 7*5*8+21 = 7(5*8+3)

So, there are 301 apples in the box.

Answer 3

To solve this problem, we need to find a number that satisfies two conditions: when divided by 2, 3, 4, 5, or 6, it leaves a remainder of 1; and when divided by 7, it leaves no remainder.

Let's start by finding the least common multiple (LCM) of 2, 3, 4, 5, and 6, since this will give us a number that has a remainder of 1 when divided by each of these numbers. The LCM of these numbers is 60.

Now, let's find a number that meets the second condition, having no remainder when divided by 7. One way to do this is to look for multiples of 60 that are also divisible by 7.

Multiples of 60: 60, 120, 180, 240, 300, 360, 420, 480, 540, 600, ...

Out of these multiples, we find that 420 is divisible by 7 without any remainder.

So, the number of apples the storekeeper has is equal to the LCM of 2, 3, 4, 5, and 6 (which is 60) plus the number that satisfies the condition for 7 (which is 420).

Therefore, the storekeeper has 60 + 420 = 480 apples.