Answer:
a) 3⁵5³.
b) 1
c) 23³
d) 41·43·53
e) 1
f) 1111
Step-by-step explanation:
The greatest common divisor of two integers is the product of their common powers of primes with greatest exponent.
For example, to find gcd of 2⁵3⁴5⁸ and 3⁶5²7⁹ we first identify the common powers of primes, these are powers of 3 and powers of 5. The greatest power of 3 that divides both integers is 3⁴ and the greatest power if 5 that divides both integers is 5², then the gcd is 3⁴5².
a) The greatest common prime powers of 3⁷5³7³ and 2²3⁵5⁹ are 3⁵ and 5³ so their gcd is 3⁵5³.
b) 11·13·17 and 2⁹3⁷5⁵7³ have no common prime powers so their gcd is 1
c) The only greatest common power of 23³ and 23⁷ is 23³, so 23³ is the gcd.
d) The numbers 41·43·53 and 41·43·53 are equal. They both divide themselves (and the greatest divisor of a positive integer is itself) then the gcd is 41·43·53
e) 3³5⁷ and 2²7² have no common prime divisors, so their gcd is 1.
f) 0 is divisible by any integer, in particular, 1111 divides 0 (1111·0=0). Then 1111 is the gcd
