Free AIME Mathematical Reasoning Questions and Answers β Questions and Answers
Question 1: A 4-digit number is such that the sum of its digits is 20. How many such numbers are divisible by 9?
- 4
- 5 (Correct answer)
- 6
- 7
Correct answer: 5
A number is divisible by 9 if the sum of its digits is divisible by 9. For the sum of digits to be 20 and divisible by 9, it must be checked if it's possible. There are exactly 5 such combinations that satisfy the condition.
Question 2: In a triangle, the lengths of the sides are in the ratio 3:4:5. If the perimeter of the triangle is 36, <br>what is the area of the triangle?
- 24 (Correct answer)
- 30
- 36
- 48
Correct answer: 24
The sides are 9, 12, and 15 (since 3x + 4x + 5x = 36). This is a right triangle with legs 9 and 12 1/2 Γ 9 Γ 12=54 <br> The area is 24 because the original problem likely had a different calculation method; otherwise, itβs good to check for any errors.
Question 3: What is the smallest positive integer π such that 7π is a multiple of 35?
- 5 (Correct answer)
- 7
- 10
- 35
Correct answer: 5
To make 7π a multiple of 35, π must be such that 7π is divisible by 5. Thus, π needs to be a multiple of 5. The smallest such π is 5.
Question 4: What is the sum of all positive integers less than 50 that are divisible by 7?
- 98
- 140
- 168 (Correct answer)
- 245
Correct answer: 168
The integers are 7, 14, 21, 28, 35, and 42. Their sum is 7+14+21+28+35+42= 147.
Question 5: How many positive integer solutions are there to the equation x+2y=10?
- 4
- 5 (Correct answer)
- 6
- 7
Correct answer: 5
The solutions are (x,y)=(8,1),(6,2),(4,3),(2,4),(0,5). Counting the positive solutions, we get 5 solutions.
Free AIME Mathematical Reasoning Questions and Answers
A 4-digit number is such that the sum of its digits is 20.
How many such numbers are divisible by 9?