john wants to send a letter to peter who lives on tesla street. John doesn’t remember the house number. However, he knows that it has 4 digits, it is a multiple of 5 and 7 and that the last digit is 0. What is the minimum number of letters that John has to send to be sure that Peter receives his letter?
The correct answer and explanation is :
To solve the problem, we need to identify the house number that meets the given criteria and determine how many letters John must send to ensure that Peter receives his letter.
Step 1: Understand the given conditions
- The house number has 4 digits.
- It is a multiple of 5 and 7.
- The last digit of the house number is 0 (since it’s a multiple of 5).
We can break this problem down into several steps.
Step 2: Identify the range of possible house numbers
Since the house number is a 4-digit number, it must lie between 1000 and 9999. Additionally, the number must end in 0, so it is a multiple of 10.
Thus, the possible numbers must be multiples of both 10 (because the last digit is 0) and 7 (since the number is divisible by 7). Therefore, the number must be a multiple of both 10 and 7.
Step 3: Find the least common multiple (LCM)
The LCM of 10 and 7 is:
$$
\text{LCM}(10, 7) = 70
$$
Thus, the number must be a multiple of 70. Now we need to find the 4-digit numbers that are multiples of 70 and end in 0.
Step 4: Find the valid multiples of 70
The 4-digit numbers that are multiples of 70 can be found by considering the range from 1000 to 9999. We can find the smallest and largest multiples of 70 within this range.
- The smallest multiple of 70 greater than or equal to 1000 is:
$$
\left\lceil \frac{1000}{70} \right\rceil = 15 \quad \text{so} \quad 15 \times 70 = 1050
$$
- The largest multiple of 70 less than or equal to 9999 is:
$$
\left\lfloor \frac{9999}{70} \right\rfloor = 142 \quad \text{so} \quad 142 \times 70 = 9940
$$
Thus, the multiples of 70 between 1000 and 9999 are:
$$
1050, 1120, 1190, \dots, 9940
$$
These numbers form an arithmetic sequence where:
- The first term $a_1 = 1050$,
- The common difference $d = 70$,
- The last term $a_n = 9940$.
Step 5: Calculate the total number of possible house numbers
To determine the total number of terms in this sequence, we use the formula for the $n$-th term of an arithmetic sequence:
$$
a_n = a_1 + (n – 1) \times d
$$
Substituting the values:
$$
9940 = 1050 + (n – 1) \times 70
$$
Solving for $n$:
$$
9940 – 1050 = (n – 1) \times 70
$$
$$
8890 = (n – 1) \times 70
$$
$$
n – 1 = \frac{8890}{70} = 127
$$
$$
n = 128
$$
Thus, there are 128 possible house numbers.
Step 6: Conclusion
John must send at least 128 letters to ensure that Peter receives his letter. This is because there are 128 possible house numbers, and sending one letter to each of these addresses guarantees that the correct one will be included. Therefore, the minimum number of letters that John has to send is 128.