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MATH 4: ASSIGNMENT 5
OCT 19, 2008
Topics discussed in class
Notation. From now on, we will usually write jsut a dot instead of multiplication sign, e.g. 5 · 3
instead of 5 × 3.
Finding factors from prime factorization. Given a prime factorization of a number, we can
easily get all factors (or divisors) of this number: all of them are obtained by removing some of the
factors from the prime factorization. For example, for the number 70 = 2 · 5 · 7, the factors are 1,
2, 5, 7, 2 · 5, 2 · 7, 5 · 7, 2 · 5 · 7.
GCD and LCM from prime factorization. To find the greatest common divisor (GCD) of two
numbers, write prime factorization for each of them, then find the factors whch appear in both
numbers and take the product of these common factors. For example, for
110 = 2 · 5 · 11
30 = 2 · 3 · 5
the GCD is 2 × 5 = 10.
To find the Least Common Multiple (LCM) of two numbers, write prime factorization for each
of them, then find the factors which appear in at least one of the numbers and take the product
of these common factors. For example, the LCM of 110 and 30 is 2 · 3 · 5 · 11 = 330.
This rule also works if one of the prime factors appears more than once: for
90 = 2 · 3 · 3 · 5
84 = 2 · 2 · 3 · 7
the GCD is 2 · 3, and LCM is 2 · 2 · 3 · 3 · 5 · 7.
Homework
Reminder: the homework is to be written on a separate sheet of paper and handed
in. Do not forget to write your name on top — and write expalanations, not just
answers!!!
1. Find prime factorizations of each of the following numbers: 256; 315; 365; 192; 30
2. Find all divisors of 315; of 64
3. Find the LCM and GCD of the following numbers
• 196 and 21;
• 365 and 30;
• 48 and 60;
4. In some school, every lesson is 45 minutes long, with three minute break between lessons.
The first lesson starts at 8:00am. When will be the next lesson that starts on an hour sharp
(i.e. at some hour and 00 minutes)?
5. Ages of Amanda, Sara, and Carly are prime numbers. Carly’s age is the sum of ages of Sara
and Amanda. Amanda is the youngest. How old is Amanda?
6. Consider the number 5 × 5 × 2 × 2 × 2 × 7 × 11. Do you think it is a multiple of 10? of
100? of 1000? In how many zeroes does it end? [hint: try doing it without performing the
multiplication]
7. Consider the product of all numbers from 1 to 25: 1 × 2 × · · · × 24 × 25. It is a very large
number so do not try to compute it. However, even without computing it you can answer
the following questions
(a) How many 2s will there be in the prime factorization for this number?
(b) How many 5s will there be in the prime factorization for this number?
(c) In how many zeroes will it end?
8. From Intensive Practice 4A workbook: P. 28, Problem 12; p. 30, problem 3. Please do these
problems on a separate sheet of paper and hand the solutions as part of your homework do not write it in the workbook itself!