What is the number of even factors of 36000 which are divisible by 9 b...
36000=2^5 * 3^2 * 5^3
Since we are talking of even factors, there must be at least one 2 in the required factors.
Since the number is divisible by 9, we must have both the threes.
We cannot have more than 1 two as it will make the number divisible by 36.
So we have 1 way of choosing 2, 1 way of choosing 3, 4 ways of choosing 5.
Thus the required number of factors are
1*1*4 = 4
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What is the number of even factors of 36000 which are divisible by 9 b...
Solution:
Given, the number is 36000.
Prime factorisation of 36000 is 36000 = 2^5 × 3^2 × 5^3
To find the number of even factors of 36000, we need to consider only the even powers of 2, 3 and 5.
- Number of even factors of 2 = 6
- Number of even factors of 3 = 3
- Number of even factors of 5 = 4
Therefore, the total number of even factors of 36000 = 6 × 3 × 4 = 72.
Now, we need to find the number of factors of 36000 that are divisible by 9 but not by 36.
- Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
- Divisibility by 36: A number is divisible by 36 if it is divisible by both 9 and 4.
To find the factors of 36000 that are divisible by 9 but not by 36, we need to consider the factors of 36000 that have exactly one factor of 3 and no more than one factor of 2^2. This is because a factor of 36000 that has two factors of 3 or two or more factors of 2^2 will be divisible by 36.
- Number of factors of 36000 with exactly one factor of 3 = 2 × 4 × 5 = 40 (2 choices for the powers of 2, 4 choices for the powers of 5, and only 1 choice for the power of 3).
- Number of factors of 36000 with no more than one factor of 2^2 = 3 × 2 × 4 = 24 (3 choices for the powers of 2, 2 choices for the powers of 5, and 2 choices for the power of 2^2).
Therefore, the number of even factors of 36000 that are divisible by 9 but not by 36 = Number of factors of 36000 with exactly one factor of 3 and no more than one factor of 2^2 = 24.
Hence, the correct answer is option B (4).
What is the number of even factors of 36000 which are divisible by 9 b...
A)20