Question

John wants to send a letter to peter who lives on tesla street john does not remember the house number however he knows that it has 4 digits it is a multiple of 5 and 7 and that 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

Answer 1

Answer: 137 letters.

Step-by-step 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.