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In which of the pairs of the following numbers is none of the two numbers a prime number, but the two numbers are co-primes?
(11, 17)
(19, 27)
(8, 25)
(21, 35)
(8, 25)
Check if both numbers have a common factor greater than 1. If not, they are co-prime. Then check if any number in the pair is prime. Eliminate pairs with any prime number.
Four pairs of numbers: (11, 17), (19, 27), (8, 25), and (21, 35).
GCD(a,b)=1⟹co-prime, prime number=divisible only by 1 and itself
Check if both numbers have a common factor greater than 1. If not, they are co-prime. Then check if any number in the pair is prime. Eliminate pairs with any prime number.
Confusing 'co-primes' (numbers whose GCD is 1) with 'prime numbers' (numbers that have no factors other than 1 and itself).
Analyze (11, 17)
Both 11 and 17 are prime numbers. This fails the condition 'none of the two numbers a prime number'.
11∈P,17∈P
Analyze (19, 27)
19 is a prime number. This fails the condition 'none of the two numbers a prime number'.
19∈P
Analyze (8, 25)
8 is composite (23) and 25 is composite (52). Checking co-prime condition: factors of 8 are {1, 2, 4, 8}, factors of 25 are {1, 5, 25}. The only common factor is 1. They are co-primes.
GCD(8,25)=1
Analyze (21, 35)
21 is composite (3×7) and 35 is composite (5×7). They share a common factor of 7, so they are not co-primes.
GCD(21,35)=7=1
C is correct because (8, 25) is the only pair where both numbers are composite and they share no common factors other than 1.
Understanding the difference between prime, composite, and co-prime is fundamental for solving questions involving divisibility and HCF/LCM.
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