Question

How many integers in the range 1..22 have inverses mod 23?

Answer #1

How many incongruent solutions does
x3 ≡ 1 have modulo 59? mod 61?
x3 ≡ 8 mod 59 and 61? How about the congruences
y5 ≡ 1 and y5 ≡ 32 mod 59 and 61?
Explain the results you get and why they are the case.

how many
ordered pairs of integers (a,b) are needed to guarantee that there
are two ordered pairs (a1,b1) and (a2,b2) such that a1=a2 (mod 9)
and b1=b2 (mod 12)

How many chromosomes are present in secondary spermatocyte?
A) 22 Pairs
B) 23
C) 23 Pairs
D) 46

a. Show that if a has a multiplicative inverse modulo N,then
this inverse is unique (modulo N).
b. How many integers modulo 113 have inverses? (Note: 113 =
1331.)
c. Show that if a ≡ b (mod N) and if M divides N then a ≡b (mod
M).

1) Compute 3^23 mod 5 by using modular arithmetic. List the
steps.
2) S4 is a group of all permutations of 4 distinct symbols. List
all the elements of S4.
Is S4 an abelian group? Use an example to explain.
3) φ(25) refers to the positive integers less than 25 and
relatively prime to 25. List
these integers.

Recall that we have proven that we have the relation “ mod n” on
the integers where a ≡ b mod n if n | b − a . We call the set of
equivalence classes: Z/nZ. Show that addition and multiplication
are well-defined on the equivalence classes by showing
(a) that you have a definition of addition and multiplication
for pairs which matches your intuition and
(b) that if you choose different representatives when you add or
multiply, the...

a) How many positive integers are divisors of 243,000,000? b)
How many positive integers divide both 243,000,000 and
1,440,000

how many positive integers less than 1000 have no
repeated digits?

How many of the integers from 1 to 200 are divisible by 3,4 or
5?

Given an array, A, of n−2 unique integers in the range from 1 to
n, describe an O(n)-time method for finding the two integers in the
range from 1 to n that are not in A. You may use only O(1) space in
addition to the space used by A.

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