Answer: 137 letters.
Explanation:
The data is:
The number has 4 digits.
The number is a multiple of 5
The number is a multiple of 7
The last digit is a 0.
Then, let's find all the possible numbers that met these conditions.
Our number can be formed with 4 digits, A, B, C and D.
D is the last one, so we have D = 0.
So the number is: ABC0
We know that all the numbers that end with 0 are multiples of 5, so that condition does not matter.
Now, a number is divisible by 7 if:
Take the last digit of the number, double it and subtract it to the remaining number, if the result is multiple of 7, then the initial number is a multiple of 7.
In this case the last digit is 0, so when we double it and subtract it, we actually are not changing the other 3 digits.
Then we have that:
ABC must be a multiple of 7.
So now our problem reduces to find the number of 3-digit numbers that are multiples of 7.
The first multiple of 7 that has 3 digits is:
7*15 = 105.
And the largest possible 3 digit number is 999, now let's try to divide it by 7 to see if this is also a multiple of 7.
999/7 = 142.7
the quotient is not integer, so 999 is not a multiple of 7.
The next option is 998, we can do the same here:
998/7 = 142.6
And so on, we will find that the largest 3 digit multiple of 7 is:
994, such that:
7*152=994.
Then we have:
Smallest: 7*15 = 105
Largest: 7*152 = 994.
Then the number of multiples of 7 between 999 and 100 will be equal to the difference between the quotients between the multples and 7.
105/7 = 15
994/7 = 152
152 - 15 = 137
So we have 137 multiples of 7 with 3 digits.
Then we have 137 4-digit numbers, that end with a zero and also are divisible by 5 and 7.
So John must send 137 letters if he wants to be shure that Peter will get the card.