# Word Document

David Rock’s House Number

David's house has a number.

1) If the house number is a multiple of 3, then it is a number from 50 to 59

2) If the house number is NOT a multiple of 4, then it is a number from 60 to 69

3) If the house number is NOT a multiple of 6, then it is a number from 70 to 79

What is the house number? Note: Conditions 1), 2), and 3) must all be satisfied.

Solution:

1) If the house number is a multiple of 3, then it is a number from 50 to 59

The only multiples of 3 between 50 and 59 are 51, 54, and 57.

2) If the house number is NOT a multiple of 4, then it is a number from 60 to 69

51, 54, 57 are not multiples of 4 and are not between 60 and 69. Thus, the house number is not 51, 54, or 57, and is therefore not a multiple of 3. Since the house number is not a multiple of 3, we know it is also not a multiple of 6. Thus, the number must be between 70 and 79.

3) If the house number is NOT a multiple of 6, then it is a number from 70 to 79

70, 71, 72, 73, 74, 75, 76, 77, 78, 79 are our options now. 72, 75, and 78 are multiples of 3, which we know the house number is not. We are left with:

70, 71, 73, 74, 76, 77, 79. These numbers satisfy parts (1) and (3). Now, we need one that is a multiple of 4 since these are not included in 60 to 69. The only multiple of 4 in this list is 76.

Will the problem have a unique solution if the interval in 3) is changed from 80 to 89?

If we look at the list of numbers between 80 and 89 that are not multiples of 3 or 6, we have 80, 82, 83, 85, 86, 88, and 89. Both 80 and 88 are multiples of 4. Thus the solution would not be unique under these conditions.