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Let try THIS!! part 2

Labels: | at 2:03 AM

1)Smart Kid Kindergarten had organized their mega gathering with all their branches all over Malaysia. Each branch was grouped together and will divide equally but some of the kids will help their teacher. For first game, 5 kids help the teachers while others were dividing into 6 equally. For second game, 4 kids help the teacher, and others divide into 11 equally. For third game, 3 kids help the teacher and others 17 equally. Thus, what is the least number of kids in the gathering?

2)A band of 17 pirates stole a sack of gold coins. When they tried to divide the fortune into equal portions, 3 coins remained. In the ensuing brawl over who should get the extra coins, one pirate was killed. The wealth redistributed, but this time an equal division left 10 coins. Again an argument develop, another pirate was killed. But now the total was evenly distributed among the pirates. What was the least number of gold coins that could have been stolen?


hint;
~Chinese Remainder Theorem
~Euclidean Algorithm
~Linear Combination

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GooDLucK every1

| at 7:22 PM

For all classmates,friends,students of IIUM kUantan.,.
Goodluck for midTerms,n quizzes,.Do the Best!!

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GAUSS HISTORY

Labels: | at 8:48 PM

German Mathematician, Carl Friedrich Gauss (1777-1855) had discover another approach to divisibility questions is through the arithmetic of remainder, or commonly known as the theory of congruence. In 1801, when Gauss was 24 years old, he introduced the foundation of modern number theory in his book Disquisitiones Arithmeticae.

Gauss was one of those remarkable infant prodigies. As a child of age three, he corrected an error in his father’s payroll calculations. His arithmetical powers so overwhelmed his schoolmasters, that by the time, Gauss was 7 years old, they admitted nothing more they could teach the boy. It is said that in his first arithmetic class Gauss astonished his teacher by instantly solving what is intended to be a “busy work” problem. Find the sum of all numbers from 1 to 100. The young Gauss later confessed to having recognized the pattern,

1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, …,50 +51 = 101.

Because there are 50 pairs of numbers each of which adds up to 101, the sum of all number must be 50*101=5050.This technique provides another way of deriving the formula
1+2+3+…+ n = n(n=1)
                            2
The most extraordinary achievement of Gauss was more in the realm theoretical astronomy than of mathematics. From the scanty data available, Gauss was able to calculate the orbit of Ceres with amazing accuracy, and the elusive planet was rediscovered at the end of the year in almost exactly the position he had forecasted. The success brought Gauss worldwide fame. Although Gauss adorned every branch of mathematics, he always held number theory in high esteem and affection.

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Example Of Diophantine Equation

Labels: | at 7:13 PM

A grocer sells a 1-gallon container milk for 79 cents and a ½ gallon container of milk for 41 cents. At the end of day he sold $63.58 worth of milk. How many 1 gallon and ½ gallon container that he sell?
Let assume x as 1 gallon container and y as ½ gallon container.
79x + 41 = 6358
So,we need to find the gcd (79,41).Using Euclidean Algorithm,the calculation are as follow:
79 = 1 x 41 + 38
41 = 1 x 38 + 3
38 = 12 x 3 +2
3 = 1 x 2 + 1
2 = 2 x 1+ 0
So,the gcd (79,41) is 1
Since gcd (79,41) =1 and 1 6358 ,so the equation 79x + 41x =6358 has a solution.
To obtain 1 as linear combination of 79 and 41,we work back through the calculation as follow:
1 = 3 - 1(2)
= 3 - 1(38 – 12(3))
= -1(38) + 13(3)
= -1(38) + 13(41 – 1(38))
= -14(38) + 13(41)
= -14(79 – 1(41)) + 13(41)
= -14(79) + 27(41)
Upon multiplying the calculation by 6358,(because 6358 x 1=6358),so we get
6358 = -89012 (79) + 171666 (41)
Thus,we get Xo = -89012 and Yo = 171666 is the solution for 79x + 41y = 6358
All other solution are,
x = -89012 + 41t ≥ 0
41t ≥ 89012
t ≥ 2171
y = 171666 – 79t ≥ 0
- 79t ≥ - 171666
t ≤ 2173
Thus,we can take 2172 as t and put in into the equation,
x = -89012 + 41(2172) =40
y=171666 - 79(2172) =78
Thus,we get x = 40 and y = 78,
means 40 containers for 1 gallon milk and 78 containers of ½ gallon milk sold for $63.58.

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