Download Author: Least Common Multiple Group Members: 1. (a) Find the first

Author:
Least Common Multiple
Group Members:
1. (a) Find the first eight positive multiples of 315.
(b) Find the first eight positive multiples of 189.
(c) Find the smallest positive number that is a multiple of both 315 and 189.
2. (a) Find the prime factorization of 315.
(b) Without doing any computations determine whether or not 32 · 5 · 7 · 8 is a multiple of 315.
(c) Without doing any computations determine whether or not 32 · 5 · 7 · 19 is a multiple of 315.
(d) Without doing any computations determine whether or not 2 · 32 · 5 · 7 · 19 is a multiple of 315.
(e) Without doing any computations determine whether or not 32 · 52 · 73 is a multiple of 315.
(f) Without doing any computations determine whether or not 3 · 5 · 7 · 112 is a multiple of 315.
(g) Without doing any computations determine whether or not 32 · 72 is a multiple of 315.
(h) Explain how you determined your answers to the above questions. You should not need to explain each
one separately; the explanation you put here should work for every problem above.
3. Find the prime factorization of 1800.
4. Find the prime factorization of 2646.
5. Match each of the following numbers with the category it belongs in.
22 · 33 · 52 · 72 = 132300
This number is a multiple of both 1800 and 2646
23 · 34 · 52 · 72 · 11 = 8731800
This number is a multiple of 1800 but not a multiple of 2646
24 · 33 · 52 · 7 = 75600
This number is a multiple of 2646 but not a multiple of 1800
2 · 32 · 52 · 72 = 22050
This number is not a multiple of 1800 nor 2646
6. Suppose I have a number that is a multiple of 1800 and 2646.
(a) Could my number have exactly one 2 in its prime factorization?
(b) Could my number have exactly two 2’s in its prime factorization?
(c) Could my number have exactly three 2’s in its prime factorization?
(d) Could my number have exactly four 2’s in its prime factorization?
(e) Could my number have more than four 2’s in its prime factorization?
(f) Could my number have exactly one 11 in its prime factorization?
(g) Could my number have more than one 11 in its prime factorization?
(h) What is the smallest amount of 3’s that my number could have in its prime factorization?
(i) What is the smallest amount of 5’s that my number could have in its prime factorization?
(j) What is the smallest amount of 7’s that my number could have in its prime factorization?
(k) What is the smallest amount of 11’s that my number could have in its prime factorization?
(l) Given any other prime (i.e. other than 2,3,5,7,11), what is the smallest amount of times that prime
could show up in the prime factorization of my number?
(m) If I told you I have the smallest number that is a multiple of 1800 and 2646, what’s my number?
7. Use the prime factorization method to find the least common multiple of the following numbers.
(a) 1274 and 16660
(b) 7605 and 35625