How many positive integers, a, are there such that 2027 divided by a leaves a remainder of 7?
the answer is 8
12
To find the number of positive integers that satisfy the given condition, we need to determine the divisors of 2020.
Step 1: Find the prime factorization of 2020.
To find the prime factorization of 2020, we need to find the prime numbers that divide it.
2020 ÷ 2 = 1010
1010 ÷ 2 = 505
505 ÷ 5 = 101
The prime factorization of 2020 is 2 × 2 × 5 × 101.
Step 2: Find the divisors of 2020.
To find the divisors, we need to consider all possible combinations of multiplying the prime factors.
The divisors of 2020 are:
1, 2, 4, 5, 10, 20, 101, 202, 404, 505, 1010, and 2020.
Step 3: Determine the number of divisors that satisfy the condition.
Since we want to find the positive integers (a) that leave a remainder of 7 when dividing 2027, we need to check which divisors satisfy this condition.
Let's calculate the remainders when dividing 2027 by each of the divisors:
2027 ÷ 1 = 2027 (remainder: 0)
2027 ÷ 2 = 1013 (remainder: 1)
2027 ÷ 4 = 506 (remainder: 3)
2027 ÷ 5 = 405 (remainder: 2)
2027 ÷ 10 = 202 (remainder: 7)
...
and so on.
From the above calculations, we can see that when dividing 2027 by the number 10, we get a remainder of 7. Therefore, the only positive integer (a) that satisfies the condition is 10.
In conclusion, there is only 1 positive integer (a) such that when dividing 2027 by (a), a remainder of 7 is obtained.