What is a single digit number that is congruent to (1234)$^{64}$ + 3,333,333,333,333 (mod 10)?
Show all work.
2026-03-27 05:38:52.1774589932
Congruency of Numbers: Last digit of $(1234)^{64} + 3,333,333,333,333$
148 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in ELEMENTARY-NUMBER-THEORY
- Maximum number of guaranteed coins to get in a "30 coins in 3 boxes" puzzle
- Interesting number theoretical game
- How do I show that if $\boldsymbol{a_1 a_2 a_3\cdots a_n \mid k}$ then each variable divides $\boldsymbol k $?
- Using only the digits 2,3,9, how many six-digit numbers can be formed which are divisible by 6?
- Algebra Proof including relative primes.
- How do I show that any natural number of this expression is a natural linear combination?
- Counting the number of solutions of the congruence $x^k\equiv h$ (mod q)
- algebraic integers of $x^4 -10x^2 +1$
- What exactly is the definition of Carmichael numbers?
- Number of divisors 888,888.
Related Questions in MODULAR-ARITHMETIC
- How do I find the least x that satisfies this congruence properties?
- Counting the number of solutions of the congruence $x^k\equiv h$ (mod q)
- Remainder of $22!$ upon division with $23$?
- Does increasing the modulo decrease collisions?
- Congruence equation ...
- Reducing products in modular arithmetic
- Product of sums of all subsets mod $k$?
- Lack of clarity over modular arithmetic notation
- How to prove infinitely many integer triples $x,y,z$ such that $x^2 + y^2 + z^2$ is divisible by $(x + y +z)$
- Can $\mathbb{Z}_2$ be constructed as the closure of $4\mathbb{Z}+1$?
Related Questions in CONGRUENCES
- How do I find the least x that satisfies this congruence properties?
- Counting the number of solutions of the congruence $x^k\equiv h$ (mod q)
- Considering a prime $p$ of the form $4k+3$. Show that for any pair of integers $(a,b)$, we can get $k,l$ having these properties
- Congruence equation ...
- Reducing products in modular arithmetic
- Can you apply CRT to the congruence $84x ≡ 68$ $(mod$ $400)$?
- Solving a linear system of congruences
- Computing admissible integers for the Atanassov-Halton sequence
- How to prove the congruency of these triangles
- Proof congruence identity modulo $p$: $2^2\cdot4^2\cdot\dots\cdot(p-3)^2\cdot(p-1)^2 \equiv (-1)^{\frac{1}{2}(p+1)}\mod{p}$
Related Questions in DECIMAL-EXPANSION
- Finding the period of decimal
- Which sets of base 10 digits have the property that, for every $n$, there is a $n$-digit number made up of these digits that is divisible by $5^n$?
- Is a irrational number still irrational when we apply some mapping to its decimal representation?
- Why the square root of any decimal number between 0 and 1 always come out to be greater than the number itself?
- Why does the decimal representation of (10^x * 10^y) always suffix the same representation in binary?
- Digit sum of $x$ consisting of only 3,4,5,6 = digit sum of $2x$
- How many 3 digits numbers are equal to the sum of their first digit plus their second digit squared plus the third cubed?
- Is it possible to determine if a number is infinitely long?
- What is the logic behind the octal to decimal conversion using the expansion method?
- Is the real number whose $n^{\rm th}$ digit after the decimal point in decimal representation is the leading digit of $2^n$ a rational number?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
HINTS: We can immediately reduce this problem to $(1234)^{64} + 3 \mod 10$ (Can you explain why?)
Then the problem is really to reduce $(1234)^{64} \mod 10$. But this is actually just $4^{64}\mod 10$. Can you see why this one happens? Can you finish from here?