"Finding the smallest n such that n2 ends..." solved and explained

Zane Decker

Answered question

2022-04-05

Finding the smallest n such that

n^{2}

ends with 00001
The problem is to find the smallest natural number n so that its square's last 5 digits are 00001. n and

n^{2}

cannot begin with 0.

Answer & Explanation

Kaitlynn Craig

Beginner2022-04-06Added 13 answers

Step 1
You want to find n such that $n^{2} \equiv 1 \pm \mod 10^{5}$
By Chinese remainder theorem, this is equivalent to
$n^{2} \equiv 1 \pm \mod 2^{5}$ and $n^{2} \equiv 1 \pm \mod 5^{5}$
We have
$n^{2} \equiv 1 \pm \mod 2^{5} \Leftrightarrow n \equiv \pm 1 \pm \mod 2^{4}$
and
$n^{2} \equiv 1 \pm \mod 5^{5} \Leftrightarrow n \equiv \pm 1 \pm \mod 5^{5} .$
Again by Chinese remainder theorem, these congruence relations give four possibilities of $n mod 5 \cdot 10^{4}$ , namely
$n \equiv 1, 18751, 31249, 49999 \pm \mod 50000 .$
From here, we see that the smallest such n is 18751.

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Pre-Algebra

Which expression has both 8 and n as factors???
One number is 2 more than 3 times another. Their sum is 22. Find the numbers
8, 14
5, 17
2, 20
4, 18
10, 12
Perform the indicated operation and simplify the result. Leave your answer in factored form
$[\frac{(4 x - 8)}{(- 3 x)}] . [\frac{12}{(12 - 6 x)}]$
An ordered pair set is referred to as a ___?
Please, can u convert 3.16 (6 repeating) to fraction.
Write an algebraic expression for the statement '6 less than the quotient of x divided by 3 equals 2'.
A) $6 - \frac{x}{3} = 2$
B) $\frac{x}{3} - 6 = 2$
C) 3x−6=2
D) $\frac{3}{x} - 6 = 2$
Find: $2.48 \div 4$ .
Multiplication $999 \times 999$ equals.
Solve: (128÷32)÷(−4)=
A) -1
B) 2
C) -4
D) -3
What is $0.78888 ... . .$ converted into a fraction? $(0.7 \bar{8})$
The mixed fraction representation of 7/3 is...
How to write the algebraic expression given: the quotient of 5 plus d and 12 minus w?
Express 200+30+5+4100+71000 as a decimal number and find its hundredths digit.
A)235.47,7
B)235.047,4
C)235.47,4
D)234.057,7
Find four equivalent fractions of the given fraction: $\frac{6}{12}$
How to find the greatest common factor of $80 x^{3}, 30 y x^{2}$ ?