MATH 135
Solutions to
Assignment 3
W10
This assignment is due at 8:30am on Wednesday January 27, in the drop box outside the Tutorial
Centre, MC 4067
Remember to show all of your computational work.
The following list of the ﬁrst few primes may be useful:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101,
103, 107, 109, 113, 127, 131, 137, 139, .
. .
1. [
8 marks, distributed as (a) 4, (b) 2, (c) 2
]
(a) Find the prime factorization of
a
= 10! and
b
= 2
20

1.
Hint
: For
b
, recall that (
x
2

y
2
) = (
x
+
y
)(
x

y
).
Answer
:
a
= 10! = 2
8
·
3
4
·
5
2
·
7, and
b
= (2
5

1)(2
5
+ 1)(2
10
+ 1) = 31
·
33
·
1025 = 31
·
(3
·
11)(5
2
·
41) = 3
·
5
2
·
11
·
31
·
41
(b) Find the number of positive factors of
a
and
b
.
Hint
: Given the prime factorization of
x
,
x
=
p
d
1
1
p
d
2
2
p
d
3
3
. . . p
d
n
n
, the number of positive
factors of
x
is given by (
d
1
+ 1)(
d
2
+ 1)(
d
3
+ 1)
. . .
(
d
n
+ 1).
Answer
: We use the formula: thus
a
has (8+1)(4+1)(2+1)(1+1) = 270 factors, and
b
has (1 + 1)(2 + 1)(1 + 1)
3
= 48 factors
(c) Find the greatest common divisor and the least common multiple of
a
and
b
.
Answer