SOLUTION: ) Prove that for any positive integers a and b, gcd(a,b) · lcm(a,b) = a·b. b) If the product of two integers is 27 · 38 · 52 · 72 and their greatest common divisor is 23

Question 1158218: ) Prove that for any positive integers a and b, gcd(a,b) · lcm(a,b) = a·b.
b) If the product of two integers is 27
· 38
· 52
· 72
and their greatest common divisor is 23
· 34
· 5 what
is their least common multiple?
c) Evaluate the following quantities.
(i) 45 mod 8 (ii) 33 mod 7
(iii) 45 div 7 (iv) ((18 mod 14) + (− 35 mod 7)) mod 8
(v) 623456 mod 5 (vi) 412345 mod 5
(vii) Find three (integer) values for c such that 11 ≡ c mod 5
Answer by Shin123(626) (Show Source): You can put this solution on YOUR website!
The gcd of two numbers can be found by factoring the two numbers and multiplying all the common factors together. The lcm of two numbers can be found by factoring the two numbers and excluding all the common factors and multiplying them together. If you multiply the gcd by the lcm, the gcd replaces the factors excluded in the lcm, resulting in all the factors, which results in the two numbers multiplied together.
b) There are no such integers.
c) 45 mod 8 ≡ 40 mod 8 + 5 mod 8 ≡ 5 mod 8 ≡ 5. ii) 33 mod 7 = (28+5) mod 7=28 mod 7+5 mod 7= 5. iii) 45/7=6 R3. iv)((18 mod 14) +(-35 mod 7)) mod 8=(4+0) mod 8=4.
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