QUANTUM COMPUTING - EXAMPLE 32.4 : A system of linear congruences consists of 3

QUANTUM COMPUTING - EXAMPLE 32.4 : A system of linear congruences consists of 3 equations : X ≡ 1 (mod 2), X ≡ 3 (mod 3), X ≡ 4 (mod 5). X has positive values. (a)(i) List the values of these equations from 1 to approximately 40. (ii) Find the first smallest value and second smallest value of X. (iii) Guess the third smallest value of X. (b) Let X ≡ Aa (mod Ma), X ≡ Ab (mod Mb), X ≡ Ac (mod Mc). According to Chinese remainder theorem, X ≡ (Aa x Ya x Md + Ab x Yb x Me + Ac x Yc x Mf) [ mod (Ma x Mb x Mc) ]. (i) Show that Ma, Mb and Mc have the greatest common divisor of Ma x Mb x Mc. (ii) Find the values of Md, Me and Mf if Md = Mb x Mc, Me = Ma x Mc and Mf = Ma x Mb. (iii) Find the values of Ya, Yb and Yc if Ya = Remainder of (Md / Ma), Yb = Remainder of (Me / Mb) and Yc = Remainder of (Mf / Mc). (iv) Use Chinese remainder theorem to find X.

Question Posted / kangchuentat

1 Answers
2414 Views
I also Faced
E-Mail Answers

QUANTUM COMPUTING - EXAMPLE 32.4 : A system of linear congruences consists of 3 equations : X &equiv..

Answer / kangchuentat

QUANTUM COMPUTING - ANSWER 32.4 : (a)(i) 1 (mod 2) = 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39. 3 (mod 3) = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39. 4 (mod 5) = 4, 9, 14, 19, 24, 29, 34, 39. (ii) By observation on X, 3 equations have common values of 9 and 39. First smallest value = 9, second smallest value = 39. (iii) Third smallest value = second smallest value + (second smallest value - first smallest value) = 39 + (39 - 9) = 69. (b)(i) Let Ma = 2, Mb = 3, Mc = 5 where they are prime numbers. Their greatest common divisor is 2 x 3 x 5 = Ma x Mb x Mc (shown). (ii) Md = Mb x Mc = 3 x 5 = 15, Me = Ma x Mc = 2 x 5 = 10, Mf = Ma x Mb = 2 x 3 = 6. (iii) Ma = 2, Mb = 3, Mc = 5, Md = 15, Me = 10, Mf = 6. Md / Ma = 15 / 2 = 7 remain 1, Ya = 1. Me / Mb = 10 / 3 = 3 remain 1, Yb = 1. Mf / Mc = 6 / 5 = 1 remain 1, Yc = 1. (iv) Let Aa = 1, Ab = 3, Ac = 4, Ya = 1, Yb = 1, Yc = 1, Ma = 2, Mb = 3, Mc = 5, Md = 15, Me = 10, Mf = 6. X ≡ (Aa x Ya x Md + Ab x Yb x Me + Ac x Yc x Mf) [ mod (Ma x Mb x Mc) ] = (1 x 1 x 15 + 3 x 1 x 10 + 4 x 1 x 6) [ mod (2 x 3 x 5) ] = 69 mod 30 = 39 mod 30 = 9 mod 30. The answer is given by Kang Chuen Tat; PO Box 6263, Dandenong, Victoria VIC 3175, Australia; SMS +61405421706; chuentat@hotmail.com; http://kangchuentat.wordpress.com.

Is This Answer Correct ?

0 Yes

0 No

Post New Answer

More Chemical Engineering Interview Questions

X is a solid having a white colour at room temperature. It has a density about 2g/cc. Although it has melting point near 325 degree Celsius, its properties start degrading above 260 degree Celsius. The coefficient of friction is very low about 0.1. It has very good dielectric properties especially at higher radio frequencies. It has a very high bulk resistivity. It is chemically inert. It is also resistant to van der Waals force. It is hydrophobic as well as lipophobic. Creep or ‘Cold Flow’ has been observed in X.

0 Answers Ford,

A shiny brown colored element ‘x’ on heating in air becomes black in cooler name the element ‘x’and black colored compound formed?

0 Answers

How to determine the particle size distribution for a given bulk solid?

3 Answers

How can the particle size distribution be determined in a given bulk solid?

0 Answers

how can we measure entropy?

1 Answers

What is a 'saltation velocity' and how is it used in designing pneumatic conveying systems?

0 Answers

Explain why does catalytic converter used in an automobile?

0 Answers

which is more effective , a single extraction with a large volume of solvent or several small volume extractions? Explain.

2 Answers

hai....i am chandrika. if anybody having test questions for HAL,HPCL,ONGC,IOL...PLS SEND THE QUES WITH SOLVED ANSWERS TO ME TO MY MAIL ID chandrismiles@yahoo.co.uk

0 Answers HAL,

What is maximum recommend pipe velocity for dry and wet gases?

0 Answers

How are plate heat exchangers used in an ammonia refrigeration system?

1 Answers HJ, Reliance,

hi, i have completed B.tech Chemical enggneering, plz can anyone provide me GVK BIO company's interview questions. iam attending the interview for the first time..

0 Answers

For more Chemical Engineering Interview Questions Click Here