Primes and Divisibility: all problems

Which of the following numbers has the smallest prime factor?

(A) 55 (B) 57 (C) 58 (D) 59 (E) 61

What is the sum of the two smallest prime factors of 250?

How many three-digit numbers are divisible by 13?

The sum of two prime numbers is 85. What is the product of these two prime numbers?

The 7-digit numbers 74A52B1 and 326AB4C are each multiples of 3. Which of the following could be the value of C?

(A) 1 (B) 2 (C) 3 (D) 5 (E) 8

The 5-digit number $\underline{2}\,\underline{0}\,\underline{1}\,\underline{8}\,\underline{U}$ is divisible by 9. What is the remainder when this number is divided by 8?

What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50?

If $3^p + 3^4 = 90$, $2^r + 44 = 76$, and $5^3 + 6^s = 1421$, what is the product of $p$, $r$, and $s$?

For any positive integer $M$, the notation $M!$ denotes the product of the integers 1 through $M$. What is the largest integer $n$ for which $5^n$ is a factor of the sum $98! + 99! + 100!$ ?

Three members of the Euclid Middle School girls' softball team had the following conversation.

Ashley: I just realized that our uniform numbers are all 2-digit primes.

Brittany: And the sum of your two uniform numbers is the date of my birthday earlier this month.

Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month.

Ashley: And the sum of your two uniform numbers is today's date.

What number does Caitlin wear?

The number AB962C is divisible by 792 where A, B, C are digits. What is the value of A?

The digits 1, 2, 3, 4, and 5 are each used once to write a five-digit number PQRST. The three-digit number PQR is divisible by 4, the three-digit number QRS is divisible by 5, and the three-digit number RST is divisible by 3. What is P?

The base-ten representation for $19!$ is $121{,}6T5{,}100{,}40M{,}832{,}H00$, where $T$, $M$, and $H$ denote digits that are not given. What is $T + M + H$?

Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, 1357, 89, and 5 are all uphill integers, but 32, 1240, and 466 are not. How many uphill integers are divisible by 15?

What is the prime factorization of 420?

What is the prime factorization of 1001?

The number 64 has the property that it is divisible by its units digit. How many whole numbers between 10 and 50 have this property?

The number 8173ABC is divisible by 165. What is the average of all such 3 digit numbers ABC that satisfy this condition?

Ayasha, Beshkno, and Chenoa were all born after 2000. Each of them was born in a year after 2000 that is divisible by exactly one of the prime numbers 2, 3 or 5. Each of these primes is a divisor of one of the birth years. What is the least possible sum of their birth years?

Sohil is thinking of a 5 digit number such that the first 3 digits are divisible by 63, the middle 3 digits are divisible by 8, and the last 3 digits are divisible by 11. What is the largest possible number he can be thinking of?

How many permutations of 123456 are divisible by their last digit? For instance, 123456 is divisible by 6, but 561234 is not divisible by 4.