28.
If n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is
A 6 B 12 C 24 D 36 E 48
Solution:
All perfect squares are composed of pairs of primes. So, let's start by breaking 72 down into primes: 72 = 2*2*2*3*3 If we break those primes into pairs, we get: (2*2)*(3*3)*2 As we can see, there's a "dangling 2". In order to form the smallest possible perfect square that's a multiple of 72, we need to pair up that 2.
So, the smallest possible perfect square that's a multiple of 72 is: (2*2)*(3*3)*(2*2)
To find the root of that perfect square, we select one of each of our pairs of primes: 2*3*2 = 12 So, the smallest possible value of n is 12; accordingly, the largest positive integer that MUST be a factor of n is also 12.
29.
How many positive integers less than 10,000 are such that the product of their digits is 210? a) 24 b) 30 c) 48 d) 54 e) 72
Solution:
Let's start by breaking 210 down into primes: 210 = 10 * 21 = 2*5*3*7 So, to start, we know that all permutation of 2,3,5,7 will fulfill the requirements: 4! = 4*3*2 = 24 However, we have to recognize that we may be able to use 1 and another number in place of a pair of our numbers. Since each digit has to be less than or equal to 9, the only substitution we can do is to use 1*6 instead of 2*3 (2*5, the next smallest pair of primes, would give us 10). So, we can also use the digits 1,5,6,7. Same as last time, this gives us 4! = 24 possible arrangements.
So, 24+24=48 nowhere does it say we need 4 digits. So, we can also make some 3 digit numbers using 5, 6 and 7. That's 3! more possibilities.. Add another 6 to our 48...
Choose D!
30.
If Ben were to lose the championship, Mike would be the winner with a probability of 1/4, and Rob 1/3. If the probability of Ben being the winner is 1/7, what is the probability that either Mike or Rob will win the championship? 1/12 ; 1/7 ; 1/2 ; 7/12 ; 6/7
Solution:
The only conditional part is Ben losing. There's no reason to multiply your individual events by the probability of Rob/Mike losing as well. So, if you just want to add the two events: Ben losing then Rob winning = 6/7 * 1/3 = 6/21 = 2/7 Ben losing then Mike winning = 6/7 * 1/4 = 6/28 = 3/14 2/7 + 3/14 = 4/14 + 3/14 = 7/14 = ½
If n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is
A 6 B 12 C 24 D 36 E 48
Solution:
All perfect squares are composed of pairs of primes. So, let's start by breaking 72 down into primes: 72 = 2*2*2*3*3 If we break those primes into pairs, we get: (2*2)*(3*3)*2 As we can see, there's a "dangling 2". In order to form the smallest possible perfect square that's a multiple of 72, we need to pair up that 2.
So, the smallest possible perfect square that's a multiple of 72 is: (2*2)*(3*3)*(2*2)
To find the root of that perfect square, we select one of each of our pairs of primes: 2*3*2 = 12 So, the smallest possible value of n is 12; accordingly, the largest positive integer that MUST be a factor of n is also 12.
29.
How many positive integers less than 10,000 are such that the product of their digits is 210? a) 24 b) 30 c) 48 d) 54 e) 72
Solution:
Let's start by breaking 210 down into primes: 210 = 10 * 21 = 2*5*3*7 So, to start, we know that all permutation of 2,3,5,7 will fulfill the requirements: 4! = 4*3*2 = 24 However, we have to recognize that we may be able to use 1 and another number in place of a pair of our numbers. Since each digit has to be less than or equal to 9, the only substitution we can do is to use 1*6 instead of 2*3 (2*5, the next smallest pair of primes, would give us 10). So, we can also use the digits 1,5,6,7. Same as last time, this gives us 4! = 24 possible arrangements.
So, 24+24=48 nowhere does it say we need 4 digits. So, we can also make some 3 digit numbers using 5, 6 and 7. That's 3! more possibilities.. Add another 6 to our 48...
Choose D!
30.
If Ben were to lose the championship, Mike would be the winner with a probability of 1/4, and Rob 1/3. If the probability of Ben being the winner is 1/7, what is the probability that either Mike or Rob will win the championship? 1/12 ; 1/7 ; 1/2 ; 7/12 ; 6/7
Solution:
The only conditional part is Ben losing. There's no reason to multiply your individual events by the probability of Rob/Mike losing as well. So, if you just want to add the two events: Ben losing then Rob winning = 6/7 * 1/3 = 6/21 = 2/7 Ben losing then Mike winning = 6/7 * 1/4 = 6/28 = 3/14 2/7 + 3/14 = 4/14 + 3/14 = 7/14 = ½
Hello, my name is Jack Sparrow. I'm a 50 year old self-employed Pirate from the Caribbean.
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