Find the prime decomposition of 1,980. Determine how many...

Find the prime decomposition of 1,980. Determine how many positive divisors 1,980 has. Find D1,980 (the divisor set of 1980). Draw the partial order diagram of D1,980 ordered by divisibility. Find the upper-bound set of 22 and 99. Find the least upper bound of 22 and 99. Find the lower-bound set of 22 and 99. Find the greatest lower bound of 22 and 99. Find the GCD(22, 99) and LCM(22, 99). What connection do you see between parts (6), (8) and (9)?

Transcript

00:02 Hi, here for the given question, we are given some sub parts.

00:06 We need to answer them for the first one.

00:08 We need to write the prime decomposition of 1980.

00:12 So here we know that this can be written as 2 square multiplied with 3 square multiplied with 5 to the power 11.

00:21 So here the prime decomposition can be written as 1980 is equal to 2 square multiplied with 3 square multiplied with 5 times 11.

00:32 Now further for the next part, we need to write down the positive divisor of 1 ,980.

00:46 So here by multiplying the exponents of the each prime factor, we can say that the positive divisor will be 2 plus 1 multiplied with 2 plus 1 multiplied with 1 plus 1, 1 plus 1.

01:01 So this is equal to 3 multiplied with 3 multiplied with 2 multiplied with 2.

01:06 So this is equal to 36.

01:09 So here in our case, we can say that 1980 has 36 positive divisors.

01:23 Now further we need to write down the set of all divisors.

01:27 So here for the third one set of divisors is equal to 1 ,2 ,3 ,4 ,5 ,6 ,10 ,11 ,12 ,15 ,20 ,22 ,30 ,33 ,44 ,55 ,60 ,66 ,110 ,132 ,165 ,220 ,330, 660 ,990 ,1980.

02:07 So this is our required third solution.

02:11 Now for the fourth one, here in our case, we need to draw partial ordered diagram partial ordered diagram of divisor of 1980.

02:35 So here in our case, this can be further written as 1980 further can be our three main factor.

02:47 First one is 130 to 660 and 330.

02:53 So again, this has further divided into two parts 44, 22, 4, 11 and 1.

03:03 Now as we can observe that this is the divisor tree.

03:06 Now further this can be written as 1 ,1 ,10 ,165 further 1 ,1 ,10 can be written as 2 and 5 and here further 22 can also be written as 11 into 2 and here in our case.

03:25 Now further 165 can be written as 3 and 6.

03:35 Now here for 330, it can be further written as 220 again 220 here we have 10 and 20.

03:44 So here further 5 and 3 has value 1, 10 and 20 has value 1.

03:50 So here this is our required solution for the fourth one.

03:56 So this is the diagram now here for the fifth part here in our case for the fifth part.

04:05 We need to calculate the value of upper bound of upper bound of 22 and 999...

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