Question

Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. How many uphill integers are divisible by 15?

a) 4
b) 5
c) 6
d) 7

Answer 1

There are 4 uphill integers divisible by 15: 15, 45, 135, and 345. These satisfy the condition of increasing digits and divisibility by 3 and 5.

Step-by-step explanation:

To find out how many uphill integers are divisible by 15, we need to recall two key properties:

1. An integer is divisible by 15 if and only if it is divisible by both 3 and 5.
2. For a number to be divisible by 5, its last digit must be either 0 or 5. Since we are considering strictly increasing digits (uphill), the only possible ending digit for our number must then be 5, as 0 would imply a preceding digit, which contradicts the uphill property.

Next, we need to ensure the number is divisible by 3. To be divisible by 3, the sum of the digits of the number must be divisible by 3. Since our uphill integer's last digit is 5, the remaining digits must sum to a number divisible by 3.

An uphill integer that is divisible by 15 could have at most two digits before the final 5 (because the digits must be in strictly increasing order). Let's start by trying to form two-digit uphill integers ending in 5:

• 135
• 345

Both of these numbers are divisible by 15 and are uphill. For single digit plus 5, we can have:

• 15
• 45

So, there are a total of 4 uphill integers that are divisible by 15, which makes the correct answer (a) 4.