The total number of 4-digit numbers whose greatest common divisor with 54 is 2 , is ____.
Correct Answer: 3000
Solution and Explanation
(B) Total 4-digit numbers divisible by \( 2 \): \[ \frac{9000}{2} = 4500. \] (C) Total 4-digit numbers divisible by \( 6 \): \[ \frac{9000}{6} = 1500. \] (D) Numbers divisible by \( 2 \) but not \( 3 \): \[ 4500 - 1500 = 3000. \]
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Concepts Used:
Fundamental Counting Principle
Assume you have 2 pairs of shoes and 3 pairs of socks. In how many different ways can you wear them? Now, the probable ways of choosing a pair of shoes are 2, since 2 pairs of shoes are available. With any pair of shoes, any of the 3 pairs of socks can be worn at a particular time. Hence, for each pair of shoes, there are 3 choices of socks. Likewise, with 2 pairs of shoes, there are 6 choices of socks available since 2 × 3 = 6. This can be understood more apparently with the help of the following figure. Consider A1 and A2 represent the 2 pairs of shoes and B1, B2, and B3 represent the 3 pairs of socks.
In the problem stated aforesaid, we use the fundamental principle of counting to get the best outcome. The multiplication principle states that if an event A can happen in x different ways and another event B can happen in y different ways then there are x × y ways of happening of both the events simultaneously.
This principle can be used to anticipate the number of ways of happening of any number of finite events. For instance, if there are 4 events that can occur in the p, q, r, and s methods, then there are p × q × r × s methods in which these events can happen simultaneously.