Mr Tan has some amount of money. If he distributes equally to 2 persons, he left with $1.
If he distributes equally to 3 persons, he left with $2.
If he distributes equally to 4 persons, he left with $3.
If he distributes equally to 5 persons, he left with $4.
If he distributes equally to 6 persons, he left with $5.
If he distributes equally to 7 persons, he left with $6.
If he distributes equally to 8 persons, he left with $7.
If he distributes equally to 9 persons, he left with $8.
If he distributes equally to 10 persons, he left with $9.
Find what is the amount of money does Mr Tan have?
To find the amount of money Mr Tan has, we can use a systematic approach.
Let's consider the first statement: "If he distributes equally to 2 persons, he is left with $1." This means that the initial amount of money Mr Tan has is $1 more than a multiple of 2.
Now, let's look at the other statements:
1. "If he distributes equally to 3 persons, he is left with $2." This implies that the amount of money is $2 more than a multiple of 3.
2. "If he distributes equally to 4 persons, he is left with $3." This means that the amount of money is $3 more than a multiple of 4.
3. "If he distributes equally to 5 persons, he is left with $4." This implies that the amount of money is $4 more than a multiple of 5.
4. "If he distributes equally to 6 persons, he is left with $5." This means that the amount of money is $5 more than a multiple of 6.
5. "If he distributes equally to 7 persons, he is left with $6." This implies that the amount of money is $6 more than a multiple of 7.
6. "If he distributes equally to 8 persons, he is left with $7." This means that the amount of money is $7 more than a multiple of 8.
7. "If he distributes equally to 9 persons, he is left with $8." This implies that the amount of money is $8 more than a multiple of 9.
8. "If he distributes equally to 10 persons, he is left with $9." This means that the amount of money is $9 more than a multiple of 10.
We can continue this process for different numbers of persons, but it becomes clear that we need to find a number that satisfies all of these conditions. This number is called the least common multiple (LCM) of the numbers 2, 3, 4, 5, 6, 7, 8, 9, and 10.
The LCM of these numbers is 2520. Therefore, Mr Tan has $2520.
Now let's explain how to calculate the LCM using prime factorization.
1. Write the prime factorizations of each number:
- 2 = 2
- 3 = 3
- 4 = 2^2
- 5 = 5
- 6 = 2 * 3
- 7 = 7
- 8 = 2^3
- 9 = 3^2
- 10 = 2 * 5
2. Take the highest power of each prime factor that appears in any of the factorizations:
- 2^3 (highest power of 2)
- 3^2 (highest power of 3)
- 5 (highest power of 5)
- 7 (highest power of 7)
3. Multiply these highest powers together:
2^3 * 3^2 * 5 * 7 = 8 * 9 * 5 * 7 = 2520
Therefore, the LCM of 2, 3, 4, 5, 6, 7, 8, 9, and 10 is 2520, and Mr Tan has $2520.