Maxie counted her dimes,when she put them in a groups of 4 she had 2 left over, groups of 5 1 left over. What is the smallest number of dimes she could have over if she has more than 10
You are looking for the smallest number N that satisfies
N mod 4 = 2, ....(1)
N mod 5 = 1 .....(2)
Since N mod 4 = 2 can only be satisfied by an even number, N can only be a number that ends in the digit 6 if (2) is to be satisfied.
Since it is over 10, and 16 does not satisfy (1), we choose 26 as the smallest number over 10 that satisfies the two given conditions.
For a formal treatment of this class of problems, look up the "Chinese remainder theorem".
To find the smallest number of dimes that Maxie could have, we need to find the least common multiple (LCM) of 4 and 5.
Step 1: List the multiples of each number:
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, ...
Step 2: Identify the smallest common multiple from the lists above:
The least common multiple of 4 and 5 is 20.
Step 3: Add 1 to the LCM to find the smallest number of dimes:
Maxie would have 21 dimes.
Therefore, the smallest number of dimes Maxie could have is 21.
To find the smallest number of dimes Maxie could have, we need to find a number that satisfies both conditions: leaving a remainder of 2 when divided by 4, and leaving a remainder of 1 when divided by 5.
Let's start by finding the smallest number that leaves a remainder of 2 when divided by 4. The remainders when dividing by 4 are 0, 1, 2, and 3. Since we want a remainder of 2, the numbers we can start with are 2, 6, 10, 14, and so on.
Next, let's find the smallest number among these that leaves a remainder of 1 when divided by 5. The remainders when dividing by 5 are 0, 1, 2, 3, and 4. Since we want a remainder of 1, we can eliminate numbers that leave a remainder of 0, 2, 3, or 4 when divided by 5.
If we continue the pattern, we see that the smallest number with a remainder of 2 when divided by 4 and a remainder of 1 when divided by 5 is 14.
Therefore, the smallest number of dimes Maxie could have is 14.