BTGmoderatorDC wrote:Each factor of 210 is inscribed on its own plastic ball, and all of the balls are placed in a jar. If a ball is randomly selected from the jar, what is the probability that the ball is inscribed with a multiple of 42?

A. 1/16
B. 5/42
C. 1/8
D. 3/16
E. 1/4

OA C

Source: Manhattan Prep

Let's first count the no. of factors of 210.

If a no. N is written as x^a*y^b*z^c, where x, y and z are prime factors of N, and a, b and c are positive integers, then

The no. of factors of N = (a+1)(b+1)(c+1)

210 = 2^1*3^1*5^1*7^1

Thus, the no. of factors of 210 = (1+1)(1+1)(1+1)(1+1) = 16;

Let's find out how many multiples of 42 are factors of 210. For that, we must divide 210 by 42.

210/42 = 5, a prime no.

Thus, 42 and 42*5 = 210 are the only two factors of 210 what are multiples of 42.

Thus, the probability that the ball is inscribed with a multiple of 42 = 2/16 = 1/8

The correct answer: C

Hope this helps!

-Jay
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BTGmoderatorDC wrote:Each factor of 210 is inscribed on its own plastic ball, and all of the balls are placed in a jar. If a ball is randomly selected from the jar, what is the probability that the ball is inscribed with a multiple of 42?

A. 1/16
B. 5/42
C. 1/8
D. 3/16
E. 1/4

OA C

Source: Manhattan Prep

Let's break 210 into primes:

210 = 21 x 10 = 3 x 7 x 5 x 2

To find the total number of factors, we add 1 to each exponent of the prime factors and then multiply those values. Since each prime factor of 210 has an exponent of 1, we have:
(1 + 1)(1 + 1)(1 + 1)(1 + 1) = 16 total factors

There are 2 multiples of 42, and they are:

2 x 3 x 7

2 x 3 x 7 x 5

Thus, the probability is 2/16 = 1/8.

Answer: C

Hi All,

We're told that each factor of 210 is inscribed on its own plastic ball, and all of the balls are placed in a jar and a ball is randomly selected from the jar. We're asked for the probability that the ball is inscribed with a multiple of 42. This question can be solved with some basic Arithmetic and 'brute force.'

Determining the factors of 210 isn't too difficult (and there are some great 'math shortcuts' that can save you some time if you know the rules of division). The factors are:
1 and 210
2 and 105
3 and 70
5 and 42
6 and 35
7 and 30
10 and 21
14 and 15

Thus, there are 16 total factors. Looking at this list, the only ones that are MULTIPLES of 42 are 42 and 210... so 2 of the 16 = 2/16 = 1/8

Final Answer: C

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Rich