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:
Here’s the explanation to determine the minimum number of letters John needs to send:
To ensure Peter receives the letter, John must send a letter to every possible house number that fits the description he has. We need to find out how many such numbers exist.
The house number has four digits, meaning it’s a number between 1000 and 9999, inclusive.
The number is a multiple of 5 and 7. A number that is a multiple of both 5 and 7 must be a multiple of their least common multiple. Since 5 and 7 are prime numbers, their least common multiple is their product: 5 * 7 = 35. So, the house number must be a multiple of 35.
Additionally, John knows the last digit of the house number is 0. A number ending in 0 is automatically a multiple of 10.
So, the house number must be a multiple of 35 and a multiple of 10. Since 35 and 10 share a common factor (5), we find the least common multiple of 35 and 10.
LCM(35, 10) = LCM(57, 25). To find the LCM, we take the highest power of each prime factor: 2¹ * 5¹ * 7¹ = 70.
Thus, the house number must be a multiple of 70. (Note: Any multiple of 70 will end in 0, so the “last digit is 0” condition is automatically satisfied by being a multiple of 70).
We are looking for the number of multiples of 70 that are also 4-digit numbers.
The range of 4-digit numbers is from 1000 to 9999.
We need to find the smallest and largest multiples of 70 within this range.
To find the smallest multiple of 70 that is 1000 or greater, we divide 1000 by 70: 1000 / 70 ≈ 14.28. The smallest integer multiple of 70 will be 70 times the next integer, which is 15.
Smallest multiple: 70 * 15 = 1050. This is a 4-digit number and ends in 0.
To find the largest multiple of 70 that is 9999 or less, we divide 9999 by 70: 9999 / 70 ≈ 142.84. The largest integer multiple of 70 will be 70 times the largest integer less than or equal to 142.84, which is 142.
Largest multiple: 70 * 142 = 9940. This is a 4-digit number and ends in 0.
The possible house numbers are 70 * k, where k ranges from 15 to 142.
To count the number of integers from 15 to 142 inclusive, we calculate 142 – 15 + 1.
142 – 15 = 127.
127 + 1 = 128.
There are 128 possible house numbers that fit all the criteria. To be absolutely sure Peter receives the letter, John must send a letter to each of these 128 possible addresses.
The final answer is
128\boxed{128}128
.