Pre-Algebra & Number (Advanced)

Harder items at competition-adjacent depth.

Phase 2 · ACT Math

Pre-Algebra & Number: core idea

Nested operations order.

How many positive factors does $72$ have?

Hint: $72=2^3\cdot3^2$ ; add one to exponents. Answer: 12.

1. A clock shows $3{:}20$ . What is the smaller angle between its hands?
2. Find the units digit of $7^{103}$ .
3. A $40$ -gallon tank is drained at $3$ gallons/minute after first draining $4$ gallons instantly. How many minutes until empty?
4. If $\frac{a}{b}=\frac{4}{7}$ and $a+b=55$ , find $a$ .
5. A number is increased by $\frac13$ of itself to become $56$ . Find the original number.
6. What is the smallest positive integer divisible by $4$ , $6$ , and $9$ ?
7. A two-digit number has digits summing to $11$ . Reversing its digits makes it $27$ greater. Find the original number.
8. A jar is filled to $\frac34$ capacity. Removing $18$ marbles leaves it $\frac12$ full. What is its capacity?
9. The product of three consecutive integers is $210$ . Find the middle integer.