Answer:
[tex]10[/tex].
Step-by-step explanation:
See below for a proof of why all but the first digit of this [tex]N[/tex] must be "[tex]9[/tex]".
Taking that lemma as a fact, assume that there are [tex]x[/tex] digits in [tex]N[/tex] after the first digit, [tex]\text{A}[/tex]:
[tex]N = \overline{\text{A} \, \underbrace{9 \cdots 9}_{\text{$x$ digits}}}[/tex], where [tex]x[/tex] is a positive integer.
Sum of these digits:
[tex]\text{A} + 9\, x= 2021[/tex].
Since [tex]\text{A}[/tex] is a digit, it must be an integer between [tex]0[/tex] and [tex]9[/tex]. The only possible value that would ensure [tex]\text{A} + 9\, x= 2021[/tex] is [tex]\text{A} = 5[/tex] and [tex]x = 224[/tex].
Therefore:
[tex]N = \overline{5 \, \underbrace{9 \cdots 9}_{\text{$224$ digits}}}[/tex].
[tex]N + 1 = \overline{6 \, \underbrace{000 \cdots 000000}_{\text{$224$ digits}}}[/tex].
[tex]N + 2021 = 2020 + (N + 1) = \overline{6 \, \underbrace{000 \cdots 002020}_{\text{$224$ digits}}}[/tex].
Hence, the sum of the digits of [tex](N + 2021)[/tex] would be [tex]6 + 2 + 2 = 10[/tex].
Lemma: all digits of this [tex]N[/tex] other than the first digit must be "[tex]9[/tex]".
Proof:
The question assumes that [tex]N\![/tex] is the smallest positive integer whose sum of digits is [tex]2021[/tex]. Assume by contradiction that the claim is not true, such that at least one of the non-leading digits of [tex]N[/tex] is not "[tex]9[/tex]".
For example: [tex]N = \overline{(\text{A})\cdots (\text{P})(\text{B}) \cdots (\text{C})}[/tex], where [tex]\text{A}[/tex], [tex]\text{P}[/tex], [tex]\text{B}[/tex], and [tex]\text{C}[/tex] are digits. (It is easy to show that [tex]N[/tex] contains at least [tex]5[/tex] digits.) Assume that [tex]\text{B} \![/tex] is one of the non-leading non-"[tex]9[/tex]" digits.
Either of the following must be true:
[tex]\text{P}[/tex], the digit in front of [tex]\text{B}[/tex] is a "[tex]0[/tex]", or[tex]\text{P}[/tex], the digit in front of [tex]\text{B}[/tex] is not a "[tex]0[/tex]".If [tex]\text{P}[/tex], the digit in front of [tex]\text{B}[/tex], is a "[tex]0[/tex]", then let [tex]N^{\prime}[/tex] be [tex]N[/tex] with that "[tex]0\![/tex]" digit deleted: [tex]N^{\prime} :=\overline{(\text{A})\cdots (\text{B}) \cdots (\text{C})}[/tex].
The digits of [tex]N^{\prime}[/tex] would still add up to [tex]2021[/tex]:
[tex]\begin{aligned}& \text{A} + \cdots + \text{B} + \cdots + \text{C} \\ &= \text{A} + \cdots + 0 + \text{B} + \cdots + \text{C} \\ &= \text{A} + \cdots + \text{P} + \text{B} + \cdots + \text{C} \\ &= 2021\end{aligned}[/tex].
However, with one fewer digit, [tex]N^{\prime} < N[/tex]. This observation would contradict the assumption that [tex]N\![/tex] is the smallest positive integer whose digits add up to [tex]2021\![/tex].
On the other hand, if [tex]\text{P}[/tex], the digit in front of [tex]\text{B}[/tex], is not "[tex]0[/tex]", then [tex](\text{P} - 1)[/tex] would still be a digit.
Since [tex]\text{B}[/tex] is not the digit [tex]9[/tex], [tex](\text{B} + 1)[/tex] would also be a digit.
let [tex]N^{\prime}[/tex] be [tex]N[/tex] with digit [tex]\text{P}[/tex] replaced with [tex](\text{P} - 1)[/tex], and [tex]\text{B}[/tex] replaced with [tex](\text{B} + 1)[/tex]: [tex]N^{\prime} :=\overline{(\text{A})\cdots (\text{P}-1) \, (\text{B} + 1) \cdots (\text{C})}[/tex].
The digits of [tex]N^{\prime}[/tex] would still add up to [tex]2021[/tex]:
[tex]\begin{aligned}& \text{A} + \cdots + (\text{P} - 1) + (\text{B} + 1) + \cdots + \text{C} \\ &= \text{A} + \cdots + \text{P} + \text{B} + \cdots + \text{C} \\ &= 2021\end{aligned}[/tex].
However, with a smaller digit in place of [tex]\text{P}[/tex], [tex]N^{\prime} < N[/tex]. This observation would also contradict the assumption that [tex]N\![/tex] is the smallest positive integer whose digits add up to [tex]2021\![/tex].
Either way, there would be a contradiction. Hence, the claim is verified: all digits of this [tex]N[/tex] other than the first digit must be "[tex]9[/tex]".
Therefore, [tex]N[/tex] would be in the form: [tex]N = \overline{\text{A} \, \underbrace{9 \cdots 9}_{\text{many digits}}}[/tex], where [tex]\text{A}[/tex], the leading digit, could also be [tex]9[/tex].
HELPPP
Which shapes have the greatest area?
(Select all that apply.)
Answer is 1st and 3rd shape
Whose area is 4 units
Answered by Gauthmath must click thanks and mark brainliest
It is Keke’s birthday, and her parents want to surprise her with a room full of balloons. Use the Fermi process
to estimate the number of balloons needed to fill Keke’s living room. Assume the living room is a rectangular
prism with a length of 30 feet, a width of 45 feet, and a height of 11 feet. Assume the balloons are spheres
with a radius of 0.45 feet. Show your work
Answer:
im not sure sorry .
Step-by-step explanation:
Which angle in ABC has the largest measure?
A.
B.
C.
D. Cannot be determined
Answer:
< B has the largest angle measure.
as it is opposite to longest side of triangle.
simplify (4+√3)(4-√3)
Answer:
13
Step-by-step explanation:
(4+√3)(4-√3) = (4)^2-(√3)^2 =16-3 = 13
Answer:
Step-by-step explanation:
(a + b)(a -b) =a² - b²
a = 4 and b = √3
(4 + √3)(4 - √3)= 4² - (√3)²
= 16 - √3*√3
= 16 - 3
= 13
Which of the following are valid probability distributions?
Answer:
Step-by-step explanation:
The dimensions of four rectangles are given below. Which has an area different from the area of each of the other three?
To determine which of the rectangles has a different area, the areas of the four rectangles must be calculated. The rectangle with a different area is rectangle b
The dimension of the 4 rectangles are:
a. length: 4x and width: 4
b. length: 11 and width: x
c. length: 2 and width: 8x
d. length: 16x and width: 1
The area of a rectangle is:
[tex]Area = Length \times Width[/tex]
Rectangle (a)
[tex]Area = 4x \times 4[/tex]
[tex]Area = 16x[/tex]
Rectangle (b)
[tex]Area = 11 \times x[/tex]
[tex]Area = 11x[/tex]
Rectangle (c)
[tex]Area = 2 \times 8x[/tex]
[tex]Area = 16x[/tex]
Rectangle (d)
[tex]Area = 16x \times 1[/tex]
[tex]Area = 16x[/tex]
Rectangles (a), (c) and (d) have the same area (i.e. 16x) while rectangle (b) has 11x as its area.
Hence, the rectangle with a different area is rectangle (b).
Read more about areas of rectangles at:
https://brainly.com/question/14383947
PLS HELP ME ON THIS QUESTION I WILL MARK YOU AS BRAINLIEST IF YOU KNOW THE ANSWER PLS GIVE ME A STEP BY STEP EXPLANATION!!
Answer:
It should be c 4,-5
hope it helps
Answer:
c. (4,-5)
Step-by-step explanation:
Point K is currently at (4,5). If reflected about x, the x-coordinate remains 4 and the y-coordinate becomes -5. Giving the answer (4,-5)
Which graph represents a decreasing trend
Answer:
Excuse me! plz provide the graphs 2.
If the circumference of a right cylinder is tripled, the volume will increase by a factor of _____.
Answer:the answer is 9
Step-by-step explanation:
Using the graph, determine the equation of the axis of symmetry.
Answer:
x = 6
Step-by-step explanation:
The axis of symmetry passes through the vertex of the graph
It is a vertical line with equation
x = c ( c is the x- coordinate of the vertex )
vertex = (6, 1 ) with x- coordinate 6 , then
equation of axis of symmetry is x = 6
Find the equivalent exponential expression.
(4^2)4
Answer:
4^8
Step-by-step explanation:
If the second 4 is an exponent, as in (4^2)^4, then multiply the exponents.
(4^2)^4 = 4^(2 * 4) = 4^8
Find C to two decimal places and find the measurement of angle be use your calculator
Let's solve
P=8B=5Using Pythagorean theorem
[tex]\\ \sf\longmapsto H^2=P^2+B^2[/tex]
[tex]\\ \sf\longmapsto c^2=8^2+5^2[/tex]
[tex]\\ \sf\longmapsto c^2=64+25[/tex]
[tex]\\ \sf\longmapsto c^2=89[/tex]
[tex]\\ \sf\longmapsto c=\sqrt{89}[/tex]
[tex]\\ \sf\longmapsto c\approx 9.22[/tex]
Does anyone know the answer plz ?
Answer:The answer is true
Step-by-step explanation:
(I hope this helped you) :)
Please show your work
[tex]1\frac{2}{3} + 2\frac{3}{4}[/tex]
Answer:
[tex]→1 \frac{2}{3} + 2 \frac{3}{4} \\ = \frac{5}{3} + \frac{11}{4} \\ = \frac{20 + 33}{12} \\ = \frac{53}{12}\\ =\boxed{4\frac{5}{12}}✓[/tex]
53/12 is the right answer.Steps:
1 2/3 + 2 3/4 can also be written as...
5/3 + 11/4
The LCD is 12 so we make both fractions have a denominator of 12.
That gives us 20/12+33/12
Now you can add the numerators (but leave the denominator the same)
That gives us 53/12, but our simplified version would give us 4 5/12 as our final answer.
Answer:
[tex]4\frac{5}{12\\}[/tex]
Please mark brainliest if that helps!
A math teacher claims that the students' test scores are directly proportional to the amount of time spent studying. Amy studies 17 hours for a test and gets a score of 68. At this rate, how many hours would she have to study to get a score of 97?
Answer:
97/4 = 24.25 hours
Step-by-step explanation:
68 = 17x
x = 68/17 = 4
Solve A = 1/2bh for h
Answer:
2A /b = h
Step-by-step explanation:
A = 1/2 bh
Multiply each side by 2
2A = 2*1/2 bh
2A = bh
Divide each side by b
2A/b = bh/b
2A /b = h
Is 8.888... a rational number?
=================================================
Explanation:
Since the decimal repeats itself infinitely (the 8s go on forever), this is one sign that we have a rational number.
Rational numbers always have some kind of infinitely repeating pattern when we convert them to decimal form.
---------------------
Extra info (optional section)
If you want to determine what this number is equal to, then let's say that
x = 8.8888...
Multiplying both sides by 10 gets us
10x = 88.888....
Now we'll subtract the two equations
The 10x-x becomes 9x while the 88.8888...-8.8888... turns into 80
We're left with 9x = 80 which solves to x = 80/9
Therefore, 80/9 = 8.8888.... showing that we have a rational number
Any rational number is basically a fraction of two whole numbers.
please answer this!!!
Answer:
C 20
Step-by-step explanation:
use sine law
Answer:
C, 20
Step-by-step explanation:
Angle c= 30
side AC= 40
sin(30)=x/40
40sin(30)=x
x=20
A bag contains eight tiles labeled A, B, C, D, E, F, G, and H. One tile will be randomly picked.
What is the probability of picking a vowel?
Write your answer as a fraction.
Answer:
1/4
Step-by-step explanation:
Count all of the vowels (this will be your numerator):
A and E
2
Count all of the letters (this will be your denominator):
A, B, C, D, E, F, G, and H
8
Write this as a fraction:
2/8
Simplify:
2/8 = 1/4
Help me please. I don't understand geometry.
a horizontally laid standard weight pipe has a diameter of 10 inches and is 12 feet long. The pipe is filled 9 inches high with water. How many gallons of water does the pipe hold? (Round your answer to the nearest gallon.) Please help.
Answer:
d isósceles hijo muy graciosa ya no me acuerdo de nada y tu q ases hablas con el
3x - 2y = 16 and x - 2y = 4 (any method <3)
Answer: (x, y) = (6, 1)
This means x = 6 and y = 1
==========================================================
Explanation:
I'll use the substitution method to solve this system.
Let's solve the second equation for x
x-2y = 4
x = 4+2y
Then plug that into the first equation so we can solve for y.
3x - 2y = 16
3(4+2y) - 2y = 16 ... x replaced with 4+2y
12+6y - 2y = 16
12 + 4y = 16
4y = 16-12
4y = 4
y = 4/4
y = 1
Lastly, use this y value to find x
x = 4 + 2y
x = 4 + 2(1)
x = 6
The solution as an ordered pair is (x, y) = (6, 1)
---------------------------------
Checking the answer:
Plug those x,y coordinates into the first equation
3x - 2y = 16
3(6) - 2(1) = 16
18 - 2 = 16
16 = 16
That confirms the first equation. Repeat for the second equation
x - 2y = 4
6 - 2(1) = 4
6 - 2 = 4
4 = 4
Both equations are confirmed.
The data below are the test scores of the students in a math class.
97, 76, 84, 82, 90, 95, 77, 79, 80, 82, 84, 77, 100, 78, 87
Create a stem-and-leaf plot to represent the data.
how???
First, we can order the set in ascending order to get
76,77, 77, 78,79, 82, 82 ,84, 84, 87, 90, 95, 97, 100.
A stem-and-leaf plot has the tens digits on the left side, and the units digits on the right side, so this set would look like
7 | 6, 7, 7, 8, 9
8 | 2, 2, 4, 4, 7
9 | 0, 5, 7
10 | 0
14. Peter needs to buy a package of pencils to write his final exam. The store offers two packages.
Package 1: pack of 6 pencils for $1.68
Package 2: pack of 9 pencils for $2.34
Determine the cost per pencil. Which is the better deal? Round to 2 decimal places. Show all your work for full marks. (3 marks)
The circumference of a circle is 16πw⁴.
Derive an expression for the area in terms of pi and w
Answer:
A = Pi (8 w**4)**2
Step-by-step explanation:
C = 16 Pi w**4 (given)
C = 2 Pi R (stand forula)
So then find R in terms of w by settting C = C from above
2 Pi r = 16 Pi w**4
2R = 16 w**4
R = 8 w**4
SO
A = Pi R**2
A = Pi (8 w**4)**2
the Marked price of a book is rupees 160 if it is sold at 15% discount find the discount amount and selling price
answer is 15%
Step-by-step explanation:
cost prices of book =160
selling prices of book =136
now,rate of discount =loss% (as discount only leads to some kind of loss)
so CP>SP
so loss=CP-SP =160-136=24
loss%= (loss/CP)÷100
=(24/160)÷100
=15%
so rate of discount =15%
calculate the speed, in ms-¹ of vehicle A and of vehicle B
9514 1404 393
Answer:
A: 0.32 m/sB: 0.56 m/sStep-by-step explanation:
The speed is the ratio of the change in distance to the corresponding change in time.
Vehicle A moves from a position of 12 m to one of 28 m in 50 seconds, so its speed is ...
A = (28 -12)/50 m/s = 16/50 m/s = 0.32 m/s
Vehicle B moves from 0 to 28 m in 50 seconds, so its speed is ...
B = (28 m)/(50 s) = 0.56 m/s
Jenn starts out with $2000 in her bank account. On average, she spends $150 a week from her account. At that rate, how many weeks will it take her to run out of money?
Answer:
About 13 weeks
Step-by-step explanation:
HELP PLEASE!!
Which best describes the relationship between the lines
with equations -3x – 4y = 1 and -6x – 8y = 2?
A. Perpendicular
B. Neither perpendicular or parallel
C. Parallel
D. Same line
Answer:
same line
Step-by-step explanation:
its the exact same equation, but the second is multiplied by 2
HELP ME PLS I NEEDA GET THIS RIGHT PLS PLS ITS PYTHAGOREAN THEOREM
2.8 mm
2mm^2=4mm
4mm+4mm = 8mm
square root of 8mm is 2sqrt2
2sqrt 2 is 2.8 mm
