Question

What are the greatest common divisors of these pairs of integers?a. 3⁷. 5³. 7³,2ⁱⁱ.3⁵.5⁹b. 11.13.17, 2⁹.3⁷.5⁵.7³c. 23³ⁱ,23ⁱ⁷d. 41.43.53.41.43.53e. 3ⁱ³. 5 ⁱ⁷.2ⁱ².7²ⁱf. 1111,0

Answer 1

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