Explanation:
the GCF is the product of all prime factors the numbers have in common.
a prime factor is a prime number that the number can be divided by without any remains.
the prime numbers start at 2.
so : 2, 3, 5, 7, 11, ...
for two numbers x, y having the GCF of 6 means they only have 2 and 3 as prime factor in common.
x = 2×3×a
y = 2×3×b
a <> b (otherwise there would be an additional shared prime factor)
what a and b factors can be found, so that x and y are between 25 and 40 ?
2×3 × 2 = 12 no
2×3 × 3 = 18 no
2×3 × 4 = 24 no
2×3 × 5 = 30 yes
2×3 × 6 = 36 yes
2×3 × 7 = 42 no
so, to verify
30 = 2×3×5
36 = 2×2×3×3
the prime factors they have in common are therefore one time 2 and one time 3 (2×3 = 6).
correct.
so, the two numbers are 30 and 36.
and because of our verification we know and are sure.