Question
upstudy study bank question image url

From a basket of tennis balls when counted in twos there was one extra, counted in threes there were two extra, counted in fours there were three extra,
counted in fives there were four extra, counted in sixes there were five extra, but counted in sevens there were no extras. At least how many tennis balls
were there in the basket?

Ask by Turner Reed. in India
Jan 09,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The smallest number of tennis balls in the basket is 119.

Solution

To find the smallest number of tennis balls that satisfies all the given conditions, we can approach the problem using modular arithmetic.

Given Conditions:

Observing the Pattern:

Notice that in each of the first five conditions, the remainder is one less than the divisor:
This suggests that is divisible by each of the numbers 2, 3, 4, 5, and 6.

Step-by-Step Solution:

  1. Find the Least Common Multiple (LCM):
    • The LCM of 2, 3, 4, 5, and 6 is 60.
    • Therefore, for some integer .
    • So, .
  2. Incorporate the Last Condition:
    • must also be divisible by 7: .
    • Simplifying: .
    • Since , the equation becomes .
    • Multiplying both sides by 2 (the modular inverse of 4 modulo 7): , which simplifies to .
    • Therefore, for some integer .
  3. Find the Smallest :
    • Plugging back into :
    • The smallest positive occurs when :
  4. Verification:
    • remainder 1.
    • remainder 2.
    • remainder 3.
    • remainder 4.
    • remainder 5.
    • remainder 0.
All conditions are satisfied.

Conclusion:

The smallest number of tennis balls in the basket is 119.
Answer: 119

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

Mind Expander

To solve this riddle, we can translate the information into a mathematical equation. Let’s denote the number of tennis balls as .
From the clues:
The first five conditions imply that is one less than a multiple of each of those divisors. Notice that if is a common multiple of 2, 3, 4, 5, and 6, we can find by first finding the least common multiple (LCM) of these numbers.
The LCM of 2, 3, 4, 5, and 6 is 60. So, we can write:
This leads us to:
Now we also know that :
This simplifies to:
Calculating gives us 4, so we need to solve:
The multiplicative inverse of 4 modulo 7 can be found to be 2 (since ). Hence,
Substituting back into our equation for :
To find the smallest positive , set :
So, the least number of tennis balls in the basket is 119.

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy