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How many of the 100 smallest positive integers are congruent to 3 (mod 7)?
Therefore, there are 14 integers in the first 100 positive integers that are congruent to 3 modulo 7.
What is the smallest four-digit number that is divisible by both 8 and 9?
To find the smallest four-digit number divisible by both 8 and 9, calculate the least common multiple (LCM) of 8 and 9. Since 8 and 9 are coprime, their LCM is:
Now, identify the smallest four-digit number divisible by 72. The smallest four-digit number is 1000. Divide 1000 by 72:
Thus, 1008 is the smallest four-digit number divisible by both 8 and 9.
To compute the greatest common divisor (GCD), take the minimum exponent for each common prime factor in the prime factorizations.
Primes not common (7 and 11) are excluded.
Actually, among any four consecutive integers: