Question
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?
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:
-
Find the Least Common Multiple (LCM):
- The LCM of 2, 3, 4, 5, and 6 is 60.
- Therefore,
for some integer . - So,
.
-
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 .
-
-
Find the Smallest
: - Plugging
back into :
- The smallest positive
occurs when :
- Plugging
-
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
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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.