Prove the divisibility of the following numbers:
1. 25^7+5^13 by 30
2. 7^6+7^5−7^4 by 11
3. 5^5−5^4+5^3 by 7

1. Make use of prime factorizations:

Pull out a common factor of :

Since , it follows that .

2. Pull out a common factor of :

3. Pull out a common factor of :

expression #3 is not equivalent. the others are.

85.560%

step-by-step explanation:

conditional probability is defined as:

p(a|b) = p(a∩b) / p(b)

in other words, the probability that a occurs, given that b has occurred = the probability that both a and b occur / the probability that b occurs.

in this example:

p(turns 64 | turns 44) = p(turns 64 & turns 44) / p(turns 44).

if someone turns 64, they must have already turned 44, so:

p(turns 64 | turns 44) = p(turns 64) / p(turns 44)

on page 42, we see that of the original 100,000, 94944 turn 44, and 81234 turn 64.   therefore:

p(turns 64 | turns 44) = 81234 / 94944

p(turns 64 | turns 44) = 0.85560