SOLUTION: Consider the integers m=2^3*3^2*5^3*7^4*11^2*13^5*17^4 and n=2^5*7^8*19^9. 1.) Note that 2^3 is a factor of both m and n. Show that 2^3 is a factor of m+n. 2.) Determine the G

Algebra ->  Exponents -> SOLUTION: Consider the integers m=2^3*3^2*5^3*7^4*11^2*13^5*17^4 and n=2^5*7^8*19^9. 1.) Note that 2^3 is a factor of both m and n. Show that 2^3 is a factor of m+n. 2.) Determine the G      Log On


   

Question 1192332: Consider the integers m=2^3*3^2*5^3*7^4*11^2*13^5*17^4 and n=2^5*7^8*19^9.
1.) Note that 2^3 is a factor of both m and n. Show that 2^3 is a factor of m+n.
2.) Determine the GCF and LCM of m and n. (explain reasoning)
3.) Show that GCF(m,n) is a factor of LCM(m,n).
Found 2 solutions by josgarithmetic, Edwin McCravy:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Question #1
Distributive Property in the reverse direction; to factorize

Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
m=2%5E3%2A3%5E2%2A5%5E3%2A7%5E4%2A11%5E2%2A13%5E5%2A17%5E4 and n=2%5E5%2A7%5E8%2A19%5E9

%22GCF%28m%2Cn%29%22%22%22=%22%222%5E3%2A7%5E4

That's because 

a.  3 is the smaller power of the common prime factor 2.
b.  4 is the smaller power of the common prime factor 7.

%22LCM%28m%2Cn%29%22%22%22=%22%222%5E5%2A3%5E2%2A5%5E3%2A7%5E8%2A11%5E2%2A17%5E4%2A19%5E9

That's because

a.  5 is the larger power of the common prime factor 2.
b.  8 is the larger power of the common prime factor 7.
c.  all the other factors of both numbers are included.

3.) 2%5E3%2A7%5E4 is a factor of 2%5E5%2A3%5E2%2A5%5E3%2A7%5E8%2A11%5E2%2A17%5E4%2A19%5E9

because a smaller power of a prime is always a factor of a
larger power of the same prime.

Edwin