To solve this problem, we need to find the greatest number such that when we add 10 to this number, the result perfectly divides each of the numbers 225, 375, and 525. Let's go through the steps in solving this:
1. Understanding the Problem:
We need a number, let's call it [tex]\( x \)[/tex], and when [tex]\( x + 10 \)[/tex] divides 225, 375, and 525 without leaving a remainder.
2. Setting Up the Equations:
This means that:
[tex]\[
225 \mod (x + 10) = 0
\][/tex]
[tex]\[
375 \mod (x + 10) = 0
\][/tex]
[tex]\[
525 \mod (x + 10) = 0
\][/tex]
Therefore, [tex]\( x + 10 \)[/tex] must be a common divisor of 225, 375, and 525.
3. Finding the Greatest Common Divisor (GCD):
To find the value of [tex]\( x + 10 \)[/tex], we first need to find the greatest common divisor (GCD) of 225, 375, and 525.
4. Calculating the GCD:
- Start with the GCD of 225 and 375.
[tex]\[
\text{GCD}(225, 375) = 75
\][/tex]
- Next, use this result to find the GCD with 525.
[tex]\[
\text{GCD}(75, 525) = 75
\][/tex]
Thus, the greatest common divisor of 225, 375, and 525 is 75. Therefore, 75 is the value of [tex]\( x + 10 \)[/tex].
5. Solving for [tex]\( x \)[/tex]:
Now that we know [tex]\( x + 10 = 75 \)[/tex], we can solve for [tex]\( x \)[/tex]:
[tex]\[
x + 10 = 75
\][/tex]
[tex]\[
x = 75 - 10
\][/tex]
[tex]\[
x = 65
\][/tex]
Conclusion:
The greatest number [tex]\( x \)[/tex] such that [tex]\( x + 10 \)[/tex] divides 225, 375, and 525 exactly is [tex]\( \boxed{65} \)[/tex].
