For how many different pairs of positive integers (a, b), with great- est common factor 1, and with a > b, does ab = 30! ?

Step-by-step explanation:

We are find two factors of 30 if when multiplied together will give 30 with a>b

when a = 30, b = 30/30 =1 where 30>1

The first pair of positive integers is (30, 1)

If a = 15

b = 30/15 = 2 where 15< 2

One of the factors is 2 and 15. The product of 2 and 15 is 30

Also if a = 10, b = 30/10 = 3

where 10>3

Another factor is 3 and 10. The product of 3 and 10 is also 30

If a = 6, then b = 30/6 = 5.

The two pairs of positive integers are 5 and 6 where 6>5 and the product of 6 and 5 is 30.

The different pair of positive integers (a, b) are;

(30, 1), (15, 2), (10, 3) and (6, 5)

Hence there are 4 different pairs of positive integers (a,b), with greatest common factor of 1,

41*67/2=1373.5

step-by-step explanation:

i just used a calculator

b. 6

step-by-step explanation:

we are given the table representing an exponential function.

we can see from the table that, as x is increasing, the value of y is increasing in the power of 6.

i.e. the following relation is obtained,

x = 1, y = 6

x = 2, y = 36 =

x = 3, y = 216 =

x = 4, y = 1296 =

x = 5, y = 7776 =

so, the geometric series obtained is .

thus, the common ratio is given by = = 6.

hence, the common ratio is 6.