Thank you for visiting The last three digits of a number N are x25 For how many values of x can N be a square of an integer. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
Answer: 3 values
Step-by-step explanation:
We need to use some algebra to find a general way to represent squares that end in 25. Then, we algebraically manipulate until we get a way to easily find the squares.
The Algebra
If the last 2 digits of the square are 25, then we can easily guess that the original number ends in 5. Let’s use some algebra to find a general way to represent squares of numbers ending in 5.
[tex](10x+5)^2[/tex]
This is a way to represent the squares of all numbers that end in 5 (like 15, 25, and so on), where a is the number that is before the 5 (e.g., (10(2) + 5)^2 = (20 + 5)^2 = (25)^2). Let’s simplify this:
[tex](10a+5)^2 = 100a^2 + 100a + 25\\= 100(a)(a+1) + 25[/tex]
We find that each square of a number ending in 5 is the value of a times a + 1 in front of the 25. For example, 15^2 = 225, since 1*2 = 2.
We can now manually find the last digit of (a)(a+1) by doing each multiplication.
0*1 = 0
1*2 = 2
2*3 = 6
3*4 = 12
4*5 = 20
5*6 = 30
6*7 = 42
7*8 = 56
8*9 = 72
Each ending digit, or ‘x’ in the problem, is either 0, 2, or 6. This will be true even if we tested from 10 to 19 or 1000 to 1009, since the last digit’s multiplication will stay the same.
N can be a square of an integer for 3 values of x: 0, 2, and 6. If x is not one of those three, N cannot be a square number.
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Rewritten by : Jeany
