Determine if 619 is a prime number Show that the decimal number 0.213213213... c

Question

Determine if 619 is a prime number Show that the decimal number 0.213213213... can be converted to a rational number. Use Euclid's algorithm to find the GCD (10764, 2300) b) Find the LCM (10764, 2300) Find the number of whole number divisors of each given value. 1, 000, 000 210^10 Simplify: (3/4 + 1/5) + 1/6 (10^-1 + 5.10^-2 + 3.10^-1)10^3 Convert 205/10, 000 to a decimal (do not use a calculator) Convert 7/2^6 and 27/250 as decimals (do not use a calculator) Write in scientific notation. i) 413, 682, 000 ii) 0.0000000856 (do not use a calculator) Find the simple interest on $8, 500 invested for 180 days at 6.25% per annum. Find the compound interest earned over 25 years compounded quarterly on $85, 000 at 7.25% per annum. Forty - two percent of the parents of the school children in laxson school district are employed at Paloma University. If the number of parents the university is 168 how many parents are in the school district.

Explanation / Answer

(According to Chegg policy, only four questions will be answered. Please post the remaining in a separate question)

1. We know that

242 = 576

252 = 675

So the square root of 619 is between 24 and 25

Further 619 is odd, so it's factors should be odd.

So it should have a prime factor from 3 to 23.

Possible factors are 3,5,7,11,13,17,19,23

619 = 3*206+1

619 = 123*5+4

619 = 88*7+3

619 = 56*11+3

619 = 47*13+8

619 = 36*17+7

619 = 32*19+11

619 = 26*23+21

Since 619 is not divisible by any of the possible prime factors,

619 is a prime

2. 0.213213......

= 213/(1000) + 213/(1000*1000) + 213/(1000*1000*1000) + ........

= 213 [ 1/1000 + 1/10002 + 1/10003 + ..............]

Since the terms form a G.P, their sum is the sum to infinite series of a GP

which is given by a / (1-r)

a = 1/1000 r = 1/1000

=> 213 [ 1/1000 / (1-1/1000) ]

= 213 [ 1/1000 / 999 /1000 ]

= 213/999

3. (10764, 2300)

Using Euclid's algorithm,

10764 = 2300 * 4 + 1564

2300 = 1564*1 + 736

1564 = 736*2 + 92

736 = 92*8 + 0

So the GCD is 92.

4. We have for two numbers a and b,

LCM(a,b) * GCD(a,b) = a*b

=> LCM(10764,2300) * 92 = 10764 * 2300

=> LCM(10764,2300) = (10764 * 2300)/92 = 269100

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