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].
AB = 20 cm, m∠A = 30°, and m∠C = 45°. Express the number of centimeters in the length of BC in simplest radical form.
Answer:
Step-by-step explanation:
x/Sin30 = 20/sin45 multiply both sides by sin30
x = 20*sin30 / sin45
sin30 = 1/2
sin45 = 0.7071
x = 20*1/2 / 0.7071
x = 10 / 0.7071
x = 10 * sqrt(2)
The ages of Raju and Ravi are in the ratio 3:4 Four years from now the ratio of their ages will be 4:5 Find their present ages
Answer: Their present age is 12 and 16.
Step-by-step explanation:
Let the required age be 3x and 4x
so
(3x+4)/(4x+4) = 4/5
5(3x+4) = 4(4x+4)
or, 15x + 20 = 16x + 16
or, 15x - 16x = 16-20
or, -x = -4
so, x = 4
now,
3x = 3*4 = 12
4x = 4*4 = 16
The two
triangles are similar.
What is the value of x?
Enter your answer in the box
Answer:
13
Step-by-step explanation:
the scale factor is 4:1 so just divide 52 by 4.
s part of a science experiment, Natasha drops a bouncy ball from various heights, h, and observes the height to which the ball rebounds, r. The table shows the results of Natasha's experiment.
Initial height, h (in meters) Rebound height, r (in meters)
0.30 0.24
0.50 0.4
0.80 0.64
1.00 0.8
1.20 0.96
Answer: r = 0.6h
your welcome :D
Answer:
The actual correct answer is r = 0.8h
Step-by-step explanation:
You divide 0.24 ÷ 0.30 and get 0.8
Do the same thing for all the rest of the numbers but,
you must put the numbers that are on the right FIRST then divide them by the numbers on the left, all of these are the same answer which means it's
r = 0.8h
Hope this helps!
2c(u-6)+(u-6) complete the expression
Answer:
the answer to the expression is u=4
There is a group of 15 people are ordering pizza. Each person wants 2 slices and each pizza has 10 slices. How many pizzas should they order?
Answer:
3 pizzas
Step-by-step explanation:
First, we can calculate how many slices are necessary. Each person wants two slices, so for each person, we must add 2 slices. We can add 2 15 times, or multiply 2 by 15 to get 2 * 15 = 30
We now know that we need 30 slices of pizza. Next, we need to figure out how many pizzas we need given this information.
For each pizza, we can add 10 slices, as there are 10 slices of pizza.
For the first pizza, we have 10 slices
For the second, we have 10+10 = 20 slices
For the third, we have 10+10+10 = 10 * 3 = 30 slices. Perfect!
Another way of figuring out the number of pizzas we need would be to divide 30 by 10, and 30/10=3 would equal the number of pizzas. We need to figure out how many pizzas we need to have 30 slices, so we can divide here.
The Marked price, inclusive of GST, of a notebook computer is $1679.90.it is sold at a discount of 10%.
a) find its marked price before GST
b)Find the GST before discount.
c)What is the selling price of the computer
d) find the GST after the discount
e)what is the percentage decrease in the GST
Answer:i don’t know
Step-by-step explanation:i don’t know
Explain how you can write a quadratic function in a factored form that would have a vertex with an x-coordinate of 3 and two distant roots
Answer:
Step-by-step explanation:
(x-3)²-4=0
(x-3)²=4=2²
x-3=±2
x=3±2
x=3-2,3+2
x=1,5
it has two distinct roots and a vertex at x=3
Help me with this....After drawing a histogram....How do u find the class in which the median lies in a histogram....
Answer:
Get all the values of the histogram and put them in order:
For example 2,5,7,1,7,3 becomes 1,2,3,5,7,7 and then you get the middle number, which is in this case (3+5)/2=3.5
Then see which of the readings lie on 3.5.
Step-by-step explanation:
9514 1404 393
Explanation:
The histogram will have a count associated with each class. You may have to read the count from the vertical scale of the graph. If it is a histogram of relative frequencies, each class will have a fraction or percentage instead of a count.
Find the total of all counts. Determine what half that number is. If the total is odd, then round up to the nearest integer. This is the index number of the median value.
Add up the counts in the classes starting at one side of the graph, working toward the other side. (Left-to-right is the usual direction for doing this.) When you find the class that increases the total to a value equal or greater than the index number, this is the class that contains the median.
__
If the histogram is a relative frequency histogram, the class that brings the total to 50% is the one containing the median.
__
There may be cases where the total count is even, and the total of all of the classes up to a point is exactly equal to the index number you're looking for. In that case, the median is between that class and the next. Here's an example:
class 1: count 3
class 2: count 5
class 3: count 4
class 4: count 4
Total count: 3+5+4+4 = 16. Median index = 16/2 = 8
Total of classes 1 and 2 = 3+5 = 8. This means exactly half of the represented values are in class 2 or below, and half are in class 3 or above. If you had access to the actual data values, you would find the median as the average of the 8th and 9th data values (when they are sorted into increasing order). Here, the median might be estimated as the value halfway between the upper bound of class 2 and the lower bound of class 3.
__
2nd example
Consider the same histogram as above, but with a count of 3 in class 4. This brings the total count to 15, and the index of the median is 15/2 = 7.5, rounds to 8. Again, the 8th data value (median) is in class 2. There is no ambiguity in this case.
brianliest!! 10 point!! hurry pls!!
Answer:
your answer is totally correct
Help and explain pls and ty
Answer:
(g ￮ f)(3) = 23
Step-by-step explanation:
Evaluate f(3) then substitute the value obtained into g(x)
f(3) = 2(3) + 4 = 6 + 4 = 10 , then
g(10) = 3(10) - 7 = 30 - 7 = 23
HELPPP
___
Thank you
Answer:
11.0.0035
12.39×10^-4
13.)0.00031
14.59800
15.3.6×10^9
Step-by-step explanation:
which expression is equivalent to
Answer:
B.
Step-by-step explanation:
(working backwards to explain)
[tex]\sqrt{5g^5h^2} \\\\\\h\sqrt{5g^5}\\\\\\g^2h\sqrt{5g}[/tex]
A triangle has sides measuring 2 inches and 7 inches. If x represents the
length in inches of the third side, which inequality gives the range of possible
values for x?
O A. 5
OB. 2
O C. 2 sxs7
D. 5sxs9
Answer:
D
Step-by-step explanation:
Answer:
The answer is A.
Step-by-step explanation:
Question 4 Multiple Choice Worth 4 points)
(01.02 LC)
What is the solution for the equation 6x - 8 = 4x?
Answer:
Algebra
Step-by-step explanation:
(+) it's the same thing btw6x -8 = 4x
(collect like terms) meaning the numbers with x go over the " = " sign
making it = -8 = 4x -6x
the signs change when it crosses over so it becomes that
-8 = -2x
-8 ÷ -2 = 4 (cause - ÷ by - is + )
A ball is thrown straight out at 80 feet per second from an upstairs window that's 15 feet off the ground. Find the ball's horizontal distance from the window at the moment it strikes the ground.
a. 0.96825
b. Can't be found
c. 77.46
d. 6.33
e. 212.23
Answer:
Step-by-step explanation:
In order to find the horizontal distance the ball travels, we need to know first how long it took to hit the ground. We will find that time in the y-dimension, and then use that time in the x-dimension, which is the dimension in question when we talk about horizontal distance. Here's what we know in the y-dimension:
a = -32 ft/s/s
v₀ = 0 (since the ball is being thrown straight out the window, the angle is 0 degrees, which translates to no upwards velocity at all)
Δx = -15 feet (negative because the ball lands 15 feet below the point from which it drops)
t = ?? sec.
The equation we will use is the one for displacement:
Δx = [tex]v_0t+\frac{1}{2}at^2[/tex] and filling in:
[tex]-15=(0)t+\frac{1}{2}(-32)t^2[/tex] which simplifies down to
[tex]-15=-16t^2[/tex] so
[tex]t=\sqrt{\frac{-15}{-16} }[/tex] so
t = .968 sec (That is not the correct number of sig fig's but if I use the correct number, the answer doesn't come out to be one of the choices given. So I deviate from the rules a bit here out of necessity.)
Now we use that time in the x-dimension. Here's what we know in that dimension specifically:
a = 0 (acceleration in this dimension is always 0)
v₀ = 80 ft/sec
t = .968 sec
Δx = ?? feet
We use the equation for displacement again, and filling in what we know in this dimension:
Δx = [tex](80)(.968) +(0)(.968)^2[/tex] and of course the portion of that after the plus sign goes to 0, leaving us with simply:
Δx = (80)(.968)
Δx = 77.46 feet
pada hari kantin sebanyak 800 naskah kupon telah dijual,harga senaskah kupon masing masing rm 30 dan rm 50 .jumlah wang diperoleh hasil daripada jualan kupon ialah rm30000.berapa naskah kupon rm30 dan rm50 yang telah dijual?
Answer:
Step-by-step explanation:
On the day of the canteen, 800 coupons were sold, the price of each coupon was RM 30 and RM 50 respectively. The amount of money earned from the sale of coupons was RM30000. How many copies of RM30 and RM50 coupons were sold?
Let:
RM 30 = x
RM 50 = y
x + y = 800 - - - (1)
30x + 50y = 30000 - - - (2)
From (1)
x = 800 - y
Put x = 800 - y in (2)
30(800 - y) + 50y = 30000
24000 - 30y + 50y = 30000
24000 + 20y = 30000
20y = 30000 - 24000
20y = 6000
y =
what is the value of x?
When a pair of parallel lines is intersected by a transversal, then
Interior opposite angles are equal.
So, (3x + 4)° = 115°
=> 3x + 4 = 115
=> 3x = 115 - 4
=> 3x = 111
=> x = 111/3
=> x = 37
Answer:
37
Step-by-step explanation:
So, if you got two parallel line, which are crossed by another line, the conterminal angles are gonna be as big as each other.
what we get outta this explanation is
3X+4=115===> 3X=111===> X=37
What is the result of subtracting the second equation from the first?
-2x+y = 0
-7x+3y = 2
Answer:
-5x + 2y=2
Step-by-step explanation:
[tex]- 7x + 3y = 2 \\ - 2x + y = 0 \\ subtracting \: this.we \: have \\ ( - 7x \: + 3y - 2) - ( - 2x + y) \\ - 7x + 2x + 3y - y - 2 \\ - 5x + 2y - 2 \\ therefore \: we \: have \\ - 5x + 2y = 2[/tex]
help i’m so confused
Answer:
-27/7
Step-by-step explanation:
put x into the equation
which inequality represents all values of x for which the quotient below is defined? √15(x-1) ÷ √2^2
Answer:
Option D: x ≥ 1.
Step-by-step explanation:
Here we have the quotient:
[tex]\frac{\sqrt{15*(x - 1)} }{\sqrt{2*x^2} }[/tex]
Here we have two restrictions for the domain.
first, the denominator can never be zero, so x must be different than zero.
Second, the arguments of the square roots can not be negative.
The one in the denominator is always positive because we have the square of x.
So let's look at the values of x, such that:
15*(x - 1) ≥ 0
(x - 1) ≥ 0
x ≥ 1
Note that when x ≥ 1, we are also removing the problem in the denominator, then we can conclude that the fraction is defined when:
x ≥ 1.
The correct option is D
What is the slope, m, and y-intercept for the line that is plotted on the fried below…?
Which represents an exterior angle of triangle ABF?
∠BAD
∠AFE
∠CAD
∠CAB
Answer:
∠CAB
Step-by-step explanation:
Exterior angles include two corner points and one point not on the triangle. Thus, ∠CAD would be an exterior angle.
Answer:
Step-by-step explanation:
A triangle has 3 interior angles. In this case, they are
<ABF
<BFA
<FAB
An exterior angle is the supplement of one of the angles. In this case, there is only 1 angle that qualifies -- <CAB which is the supplement of BAF.
All the other choices are not supplements to any of the interior angles such as
<BAD
<AFE
<CAD
I don’t know how to do this can someone help?
Answer:
Why your question is not visible
Your photo is black screen
Step-by-step explanation:
Please Mark me brainliest
Question 4
Points 2
A girl cycled a total of 15 kilometers by making 5 trips to work. How many trips will she have to make to
cover a total of 24 kilometers? Solve using unit rates.
URGENT PLS ANSWER QUICKLY
Answer:
5 is 14cm . 14 is 2pie r. 6 is 120
Answer:
Question 5 = 14
Question 6 = 120
I hope this helped!
Solve the equation and enter the value of x below. 2 + 5(x - 9) = -18
Answer:
x=5
Step-by-step explanation:
Distributive Property[tex]2+5x-45=-18\\5x=-18+45-2\\5x=25\\x=5[/tex]
x=5
Hope this helped! Please mark brainliest :)
Find the measure of the missing angle using the triangle angle sum theorm
Answer:
31°
Step-by-step explanation:
The theorem states that all angle sums of any triangle add up to 180°.
128 + 21 = 149, and to find ?°, subtract 149 from 180 to get 31°.
How do I write 4(3x+2)-9 in written form?
Answer:
12x-1
Step-by-step explanation:
First, you would have to distribute. Would mean to multiply the 4 with every number in the parenthesis.
4(3x+2)-9
12x+8-9
Now combine like terms
8-9 = -1
The answer is 12x-1
Step-by-step explanation:
4 times 3x+2 subtracted by9
hope it helps
stay safe healthy and happy.Si "m" es el lado mayor de un triángulo, dos cuyos ángulos interiores miden 30° y 60°, entonces el perímetro del triángulo es
Answer:
[tex]perimetro = \dfrac{(3 + \sqrt{3})m}{2}[/tex]
Step-by-step explanation:
triángulo 30°-60°-90°
[tex] 1~:~\sqrt{3}~:~2 [/tex]
[tex] 1m~:~m\sqrt{3}~:~2m [/tex]
[tex] \dfrac{m}{2}~:~\dfrac{m\sqrt{3}}{2}~:~m [/tex]
[tex]perimetro = \dfrac{m}{2} + \dfrac{m\sqrt{3}}{2} + m[/tex]
[tex]perimetro = \dfrac{m}{2} + \dfrac{m\sqrt{3}}{2} + \dfrac{2m}{2}[/tex]
[tex]perimetro = \dfrac{m + m\sqrt{3} + 2m}{2}[/tex]
[tex]perimetro = \dfrac{3m + m\sqrt{3}}{2}[/tex]
[tex]perimetro = \dfrac{(3 + \sqrt{3})m}{2}[/tex]
