If c and d are integers, d is divisible by 30, and cd is divisible by 20, c must be divisible by which of the following?
a) 2
b) 4
c) 5
d) 6
e) 10
To find out which of the given divisors c must be divisible by, let's analyze the given information step by step.
We are told that d is divisible by 30. This means that d can be expressed as d = 30k, where k is an integer.
We are also told that cd is divisible by 20. So, we can write cd = 20n, where n is an integer.
Substituting the value of d in terms of k in the equation cd = 20n, we get c * 30k = 20n.
To simplify the equation, divide both sides by 10: 3c * k = 2n.
This equation tells us that 3c * k must be divisible by 2. Since 2 is a prime number, the only way for the product to be divisible by 2 is if one of the factors, 3c or k, is divisible by 2.
Now, let's analyze the options:
a) 2: If c is divisible by 2, then 3c is also divisible by 2.
b) 4: If c is divisible by 4, then 3c is also divisible by 4.
c) 5: If c is divisible by 5, then 3c is also divisible by 5.
d) 6: If c is divisible by 6, then 3c and 6 are both divisible by 2.
e) 10: If c is divisible by 10, then 3c is also divisible by 10.
So, based on the equation 3c * k = 2n, we can conclude that c must be divisible by all of the options given in the question (a, b, c, d, e).
However, if we are looking for the answer that is not a subset of the others, or the smallest possible answer, we can conclude that c must be divisible by 2, which is option a).
Therefore, the answer is (a) 2.