Let N be a number created by repeating any two identical three-digit strings. That is, N=ABCABC. For example, 123123, 023023, and 400400 are in the form.

Is it true or false that every such N is divisible by 13? If true, please explain. If not, provide a counter-example.

Answers

Answer 1

Answer:

true

Step-by-step explanation:

there is a nice rule for when a number is divisible by 13 :

starting with the lowest digit value (the last on the right) form blocks of 3 digits as numbers and give them alternating signs (starting with +).

if the last block (the last on the left) has less than 3 digits, just use the remaining ones as a number.

add all these numbers (with their associated signs).

if the number calculated by this (alternative) sum is divisible by 13 then the original number is divisible by 13.

if N = abcdefghi, then (ghi - def + abc) must be divisible by 13, so that N is also divisible by 13.

in our case, N = abcabc.

the block calculation is therefore abc - abc = 0, which is always divisible by 13.

so, any number formed like this must be divisible by 13.

FYI - a quick look at why that is :

N = abcabc =

= a×10⁵ + b×10⁴ + c×10³ + a×10² + b×10 + c =

= a×(10⁵ + 10²) + b×(10⁴ + 10) + c (10³ + 1) =

= a×100100 + b×10010 + c×1001

define d = 1001 = 77×13

and we get

a×d×100 + b×d×10 + c×d

a sum is divisible by a number, if every summed up term is divisible by that number.

since d is a factor in every term, and d is divisible by 13, so is every term and therefore the whole number.

so, every number in the form of abcabc is divisible by 13.

Answer 2

It is true that every number in the form N = ABCABC, where ABC represents a three-digit string, is divisible by 13.

Is it true or false that every such N is divisible by 13?

Let's consider the number N = ABCABC, where ABC represents a three-digit string. We can express N as follows:

N = 1000 * ABC + ABC

= 1001 * ABC

Notice that 1001 is divisible by 13, as 1001 = 13 * 77. Therefore, if ABC is any three-digit string, 1001 * ABC is divisible by 13.

Since N = 1001 * ABC, and 1001 is divisible by 13, it follows that N is also divisible by 13. This holds true for every number in the form ABCABC, where ABC is a three-digit string.

This pattern holds for any three-digit string repeated twice, ensuring that every number in the form ABCABC is divisible by 13.

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Related Questions

Answer this. Show your work. Worth 10 points. Under 4 hours. URGENT.

Answers

The correct answer would be 200 miles. If 30 miles represented 3/5ths of an inch, then 72, 50, and 18 miles would all be too small.
For example 72 miles would only represent a little over an inch, so 200 miles would be the only possibility.

Here’s the work. Hope this helps

7.30. Julie wants to give a 45 -minute speech, and she speaks 120 words per minute. Her written notes contain 500 words per page. How many pages should she prepare?

Answers

Answer:

10.8 pages must be prepared

Step-by-step explanation:

We first have to figure out how many words Julie would be able to say in 45 minutes. Since she can say 120 words per minute, she can speak 120*45 or 5400 words total.

We also know that Julie can write 500 words per page. Since she has 5400 words total, she needs to prepare 10.8 pages total (5400 divided by 500). If you are looking for a whole number answer, Julie needs at least 11 pages to write her complete speech.

Answer:

The number of pages she should prepare is 10.8 pages.

Step-by-step explanation:

Given:

Julie wants to give a 45 -minute speech, and she speaks 120 words per minute.

Her written notes contain 500 words per page.

Find:

The number of pages she should prepare

Step 1 of 1

We will set up a conversion factor:

number of pages [tex]=(45$ minutes $) \cdot \frac{120 \text { words }}{1 \text { minutes }} \cdot \frac{1 \text { pages }}{500 \text { words }}$[/tex]

[tex]$$\begin{aligned}&=\frac{45 \cdot 120}{500} \text { pages } \\&=10.8 \text { pages. }\end{aligned}$$[/tex]

2. GYMNASIUM You want to estimate the number of students in your school who think the gymnasium should be remodeled. You survey 12 students on the basketball team. Determine whether the sample is biased or unbiased. Explain. (Section 10.6)

Answers

Yes your going to get biased opinions and they would all say yes because they play basketball in the gym so obviously they want good things so they want it to be remodeled

It will be biased definitely because ur asking a sport focused group of students

In triangle ABC, AB=BC, AC=10cm. From point D, which is in the middle of AB, perpendicular bisector DE is drawn to side BC and intersects it at point E. A segment is then drawn from E to A. The perimeter of triangle ABC is equal to 40 cm. Find the perimeter of △AEC.

Answers

Answer:

Step-by-step explanation:

We are given that the perpendicular bisector of side AB of ∆ABC intersects side BC at point D.

this means that side AE=BE.

Also, we could clear; ly observe that

ΔBED≅ΔAED

( since AE=BE, side ED common, ∠BED=∠AED

so by SAS congruency, the two triangles are congruent)

Now we are given that:

the perimeter of ∆ABC is 12 cm larger than the perimeter of ∆ACD.

i.e. AB+AC+BC=AC+AD+CD+12

AB+BC=AD+CD+12

as AD=BD

this means that AD+CD=BD+CD=BC

AB+BC=BC+12

AB=12

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The perimeter of 40 cm. and lengths of two sides of ∆ABC where side AC is 10 cm gives;

The perimeter of ∆AEC is 25 centimeters

How can the perimeter of the formed ∆AEC be found?

Given:

AB = BC

AC = 10 cm.

The perimeter of ∆ABC = 40 cm.

Therefore;

∆ABC is an isosceles triangle

AB + BC + AC = 40

AB + AB + 10 = 40 by substitution property of equality

2 × AB = 40 - 10 = 30

AB = 30 ÷ 2 = 15

AB = BC = 15 cm.

Using the rule of cosines, we have;

10² = 15² + 15² - 2×15×15 × cos(A)

cos(A) = (15² + 15² - 10²) ÷ (2×15×15)

A = arccos (350 ÷ 450) ≈ 51.06°

<BED ≈ 180° - (90° + 51.06°) = 38.94°

DB = AB ÷ 2

Therefore;

DB = 15 ÷ 2 = 7.5

From the rule of sines, we have;

DE/sin(51.06°) = 7.5/sin(38.94°)

DE = 7.5/sin(38.94°) × sin(51.06°) ≈ 9.28

According to Pythagorean theorem, we have;

BE ≈ √(7.5² + 9.28²) ≈ 11.93

Similarly, we have;

AE ≈ √(7.5² + 9.28²) ≈ 11.93

EC ≈ 15 - 11.93 = 3.068

Perimeter of ∆AEC = AE + EC + AC

Which gives;

The perimeter of ∆AEC = 11.93 + 3.068 + 10 ≈ 25

The perimeter of ∆AEC = 25 cm

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What is the volume of the box if it is scaled down by a factor of 1/10?
V= 308 35/64

Answers

Answer:

explanation will answer.

Step-by-step explanation:

To convert this fraction to a decimal, just divide the numerator (35) by the denominator (64). So, 35 / 64 = 35 ÷ 64 = 0.546875. Approximated Values: 35 / 64 = 0.5 rounded to 1 significant figure. 35 / 64 = 0.55 rounded to 2 significant figures. 35 / 64 = 0.547 rounded to 3 significant figures.

Answer:

Step-by-step explanation:

1.Is 4/5 equivalent to 32/45?

2.Is 7/8 equivalent to 28/32?

3.Is 1/10 equivalent to 30/100?

4.Is 6/8 equivalent to 24/32?

5.Is 5/8 equivalent to 30/70?

6.Is 2/3 equivalent to 24/100?

7.Is 2/10 equivalent to 30/60?

8.Is 1/5 equivalent to 20/100?

9.Is 5/6 equivalent to 20/30?

10.Is 3/4 equivalent to 45/70?

Answers

When comparing fractions, you can multiply or divide to find equivalent fractions.

Number 1: If I were to multiply 5 to 45, I would have to do the same to the numerator of the fraction. 45/5 = 9. So we know now that 5 times 9 equals 45. Lets do that to the numerator also. 4 times 9 equals 36 so 4/5 times nine is 32/45.

So yes, this would be equivalent.

Number 2: 7/8 times 4 equals 28/32

Yes, this would be equivalent.

Number 3: 10 times 10 is 100, 1 times ten is NOT 30.