Questions & Answers

Question

Answers

A) 12

B) 6

C) 4

D) None of these

Answer
Verified

We know that the prime factorization is finding which prime number multiplies together to make the original number.

We will now find the prime factorization of the given number 3630 by finding the prime numbers, which are divisible by this number.

Thus, the prime factorization of the given number 3630 is \[2 \times 3 \times 5 \times {11^2}\].

We will now require such divisors, which have a remainder of 1 when divided by 4.

Finding the odd divisors from the above prime factorization of the given number 3630, we get

\[2, 3, 5, 11, {11^2}\]

We will now find the possible divisors of the given number 3630 from the above odd divisors.

1

\[1 \times 3 = 3\]

\[1 \times 5 = 5\]

\[1 \times 11 = 11\]

\[1 \times 121 = 121\]

\[3 \times 5 = 15\]

\[3 \times 11 = 33\]

\[3 \times 121 = 363\]

\[5 \times 11 = 55\]

\[5 \times 121 = 605\]

Finding the remainder by dividing each of the above possible divisors of the given number 3630 with 4, we get

1

3

1

3

1

3

1

1

3

1

Thus, the divisors of the given number are 1, 5, 121, 33, 363, 605, which leaves a remainder 1 when divided by 4.

Therefore, there are only 6 possible divisors, which leave a remainder 1 when divided by 4.