Find the greatest number such that
x172y
is divisible by 45
To find the greatest number such that x172y is divisible by 45, we need to determine the largest possible values for x and y that will make the entire number divisible by 45.
The key to determining divisibility by 45 is that the number must be divisible by both 5 and 9.
First, let's consider divisibility by 5. For a number to be divisible by 5, the last digit must be either 0 or 5. Since the last digit is y in the number x172y, let's set y=5 to satisfy this condition.
Now, we have x1725 as the number. To determine divisibility by 9, we need to calculate the sum of the digits and check if it is divisible by 9. The sum of the digits in x1725 is:
x + 1 + 7 + 2 + 5 = x + 15
In order for x1725 to be divisible by 9, the sum x + 15 must also be divisible by 9. We can try the possible values of x from 0 to 9 and check if x + 15 is divisible by 9.
When x = 0, the sum is 15 which is not divisible by 9.
When x = 1, the sum is 16 which is not divisible by 9.
When x = 2, the sum is 17 which is not divisible by 9.
When x = 3, the sum is 18 which is divisible by 9. Therefore, the greatest number that satisfies the conditions is:
x1725
So, the greatest number such that x172y is divisible by 45 is 31725.
To find the greatest number that makes the given number `x172y` divisible by 45, we need to understand the divisibility rule of 45.
The divisibility rule of 45 states that a number is divisible by 45 if it is divisible by both 9 and 5.
For a number to be divisible by 9, the sum of its digits must be divisible by 9.
Let's break down the number `x172y` into its digits:
x 1 7 2 y
To make the number divisible by 9, the sum of its digits (x+1+7+2+y) must be divisible by 9.
Now, let's analyze the divisibility rule for 5. For a number to be divisible by 5, the units digit must be either 0 or 5.
To summarize, for the number `x172y` to be divisible by 45:
1. The sum of its digits (x+1+7+2+y) must be divisible by 9.
2. The units digit (y) must be either 0 or 5.
To find the greatest value of y that satisfies these conditions, we can start by determining the possible values for y. Since y needs to be divisible by 5 and the greatest units digit is 9, the only possible values for y are 0 or 5.
Next, we can find the maximum value for the sum of the digits by assuming the remaining digits are all 9s. So, the maximum value of x+1+7+2+y would be x+1+7+2+9+9.
Since we want to find the largest possible value for `x172y`, we want to maximize the value of both x and y.
If we set y as 5, then the sum of digits would be (x+1+7+2+9+9+5) which is (28+x).
For `(28+x)` to be divisible by 9, x should be equal to 1.
Therefore, the largest possible value for `x172y` that is divisible by 45 is 117295.