TJ is thinking of a two digit number If he subtracts 5 it is a multiple of 4 If he subtracts 6 it is a multiple of 5 If he subtracts 7 it is a mutliple of 6 What is TJ's number?

Answer 1

The number is 61. Let's see this.

If we subtract 5, the number will be multiple of 4. This means that the remaining number is divisible by 4, and in order to divide the remaining number by 4, the remaining number must be an even number. So the initial number must be an odd number, because we are subtracting 5 from it, and 5 is also an odd number. (In our case we can find an even number only if we subtract from an odd number another odd number) But if we subtract 6, the number will be multiple of 5, this means that that remaining number ends in either 5 or 0. We can exclude 0, because 0 causes the initial number to be an even number, which we assumed that it is an odd number. If we subtract 7, the number is a multiple of 6. This means that the sum of two digits of the remaining number is divisible by 3.

Let's look at the two digit numbers that are multiple of 4.

We can start with 20. To make it an odd number we add 1, and 20 becomes 21.

21 - 5 = 16, 16/4 =4

21 - 6 = 15, 15/5 = 3

but 21 - 7 = 14 and 1 + 4 = 5 which can't be divisible by 3, so it can't divide by 6.

Let's try 36. 36 + 1 = 37

37 - 5 = 32, 32/4 = 8

37 - 6 = 31, can't be divided by 5.

The number 9 can be divisible by 3 , and 4 + 5 = 9, so we can look at 45. If we add 7 to 45 we'll have 52, which is an even number. So let's try 54, since 5 +4 = 9. If we add 7 to 54 we'll have 61.

Let's check 61:

61 - 5 = 56, 56/4 = 14

61 - 6 = 55, 55/ 5 = 11

61 - 7 = 54, 54/6 = 9