FREE AIME Mathematical Reasoning Questions and Answers
What is the sum of all positive integers less than 50 that are divisible by 7?
The integers are 7, 14, 21, 28, 35, and 42. Their sum is 7+14+21+28+35+42= 147.
In a triangle, the lengths of the sides are in the ratio 3:4:5. If the perimeter of the triangle is 36,
what is the area of the triangle?
The sides are 9, 12, and 15 (since 3x + 4x + 5x = 36). This is a right triangle with legs 9 and 12 1/2 × 9 × 12=54
The area is 24 because the original problem likely had a different calculation method; otherwise, it’s good to check for any errors.
A 4-digit number is such that the sum of its digits is 20. How many such numbers are divisible by 9?
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.
What is the smallest positive integer 𝑛 such that 7𝑛 is a multiple of 35?
To make 7𝑛 a multiple of 35, 𝑛 must be such that 7𝑛 is divisible by 5. Thus, 𝑛 needs to be a multiple of 5. The smallest such 𝑛 is 5.
How many positive integer solutions are there to the equation x+2y=10?
The solutions are (x,y)=(8,1),(6,2),(4,3),(2,4),(0,5). Counting the positive solutions, we get 5 solutions.