MIT6_042JS10_lec22_prob

MIT6_042JS10_lec22_prob - Problem 3 The following...

This preview shows pages 1–2. Sign up to view the full content.

Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science March 31 Prof. Albert R. Meyer revised March 30, 2010, 1426 minutes In-Class Problems Week 8, Wed. Problem 1. (a) Use the Pulverizer to ﬁnd the inverse of 13 modulo 23 in the range { 1 ,..., 22 } . (b) Use Fermat’s theorem to ﬁnd the inverse of 13 modulo 23 in the range { 1 ,..., 22 } . Problem 2. (a) Why is a number written in decimal evenly divisible by 9 if and only if the sum of its digits is a multiple of 9? Hint: 10 1 (mod 9) . (b) Take a big number, such as 37273761261. Sum the digits, where every other one is negated: 3 + ( 7) + 2 + ( 7) + 3 + ( 7) + 6 + ( 1) + 2 + ( 6) + 1 = 11 Explain why the original number is a multiple of 11 if and only if this sum is a multiple of 11.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Problem 3. The following properties of equivalence mod n follow directly from its deﬁnition and simple prop-erties of divisibility. See if you can prove them without looking up the proofs in the text. (a) If a ≡ b (mod n ) , then ac ≡ bc (mod n ) . (b) If a ≡ b (mod n ) and b ≡ c (mod n ) , then a ≡ c (mod n ) . (c) If a ≡ b (mod n ) and c ≡ d (mod n ) , then ac ≡ bd (mod n ) . (d) rem( a,n ) ≡ a (mod n ) . Creative Commons 2010, Prof. Albert R. Meyer . MIT OpenCourseWare http://ocw.mit.edu 6.042J / 18.062J Mathematics for Computer Science Spring 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
View Full Document

{[ snackBarMessage ]}

Page1 / 2

MIT6_042JS10_lec22_prob - Problem 3 The following...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online