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Congruence And Its Applications

Published in: Mathematics
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  • Sowjanya K

    • Hyderabad
    • 2 Years of Experience
    • Qualification: Post Graduate
    • Teaches: EVS, Physics, Mathematics, Chemistry
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An interesting application of congruence theory to calculate the day of the week on a particular day!

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    CONGRUENCE AND ITS APPLICATION By K Sowjanya
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    FIRST REPUBLIC DAY IS JAN 26, 1950 WHAT WAS THAT DAY ????? Mahatma Gandhi born on 2nd October 1889 What was that day O CAN WE ANSWER SUCH QUESTIONS YES, MATHEMATICS IS USEFUL
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    HOW BY USING CONGRUENCE RELATIONS Congruence is one of the most important aspects of Mathematics e know that Mathematics is a seful and powerful tool of science 00 & Society
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    CONGRUENCE neveloped by Gauss - a German mathematician
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    DEFINITION Congruence Is a statement about divisibility If an integer m O divides the difference a-b we say that a is congruent to b modulo m and is written as a b(mod m)
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    THEOREM If a = b (mod m) b = c (mod m) then a = c (mod m)
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    APPLICATIONS o Using congruence we can find out whether the number is even or odd. If (mod 2) then the number is even And if (mod 2) then the number is odd.
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    TO FIND OUT A DAY OF A GIVEN PARTICULAR DATE i) ii) For this the formula the following assumptions are used : It is applicable for the years after 1600 A.D. (Gregory modified the calendar in 1582 A.D. & 1600 is the next leap year) To Count — March is first month, April 2nd and son on. (In this way December will be the 10 th month & February will be 12 th month) iii) Seven days in a week - 'O' for Sunday , '1' for Monday in this way '6' for Saturday.
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    Now 1st we shall fix the day on 1st March 1600, Suppose the day correspond to 'a' then The day on 1st March 1601 is (a+l)(mod 7) The day on 1st March 1602 The day on 1st March 1603 IS The day on 1st March 1604 ' [(a+4)+1](mod 7) (leap year)
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    In which, In the year (1600+y) the day 'd' on 1 st March is d [a+y+ no. of leap years] (mod 7) when y 100 then the no. of leap years—=y/4 -y/ 100 +y/400 Therefore, the day on which 1st March (1600+y) is given by- [a+y + y/4 - y/ 100 + y/400] (mod 7) (1)
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    Now, we shall determine 'a' by using condition - Thursday 1st March 1973 that is 4(mod 7) (2) From (1) 4 [ a+y +y/4 -y/100+y/400] (mod 7) here 373 (=1973-1600) [a+ 373 +373/4 -373/100 -373/400] (mod 7) [ a +463] (mod 7) but (a + 463) ( a +1) (mod 7) a=3 & Wednesday was the day on 1st March 1600 A.D.
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    Thus formula is , d [3 + y+ y/4 -y/ 100 + y/400 ] (mod 7) (3) take year = N & N =100 C +D then y =IOO C +D -1600 & formula becomes for 1st March (1600+y) is, d = [ 3- 2C +D +C/4 + D/4] (mod 7) (4) Using [ r+(2.6 m - 0 2) - . + D + C/4 +D/4] (mod 7)
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    26th January 1950 here r=26, m=ll ,C=19, D=49. and our formula is, CD d ] (mod 7) Substituting the values, d -2*19+49+19/4+49/4 ] (mod 7) d (mod 7) d 81. (mod 7) 81/7 (reminder = 4) Therefore the day is ThursdayC)
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    2nd The formula is, 1889 d [r + ] (mod 7) d (mod 7) d = [24-20-36+89 +4+22] (mod 7) d (mod 7) Hence the day is ...Thursday
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    REFERENCES An Introduction to theory of numbers - Ivan Niven - I-IS. Zuckerman Prathmic Sankhya Sidhhant. - Prof, M. G. Amravatkar Shree Vidya Prakashan
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    Than/ßYotm

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