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Don't Break the Rules Trivia Quiz

In each question I'll give you an example number -- but beware! The rule is a general one and doesn't specifically relate to my example. Read on and it will become clearer when I ask you what rule can be inferred from my example.

A multiple-choice quiz by garrybl. Estimated time: 5 mins.

Author
garrybl

Time
5 mins

Type
Multiple Choice

Quiz #
388,920

Updated
Dec 03 21

# Qns
10

Difficulty
Tough

Avg Score
6 / 10

Plays
214

One at a Time - Single Page - Timed Game

Question 1 of 10

1. Take a number, for example, 7846.
Which of the following must be true about any number whose last digit is 6?
Hint

It must be divisible by 6

It must be divisible by 7

It cannot be divisible by 11

It cannot be prime

NEXT>

Question 2 of 10

2. Take a number, for example 46338. The digits add up to 24. What can be said about any number whose digits add to 24? Hint

It is an even number

It is divisible by 9

It is divisible by 7

It is divisible by 3

NEXT>

Question 3 of 10

3. Take the number 45681. The digits add to 24. As it stands the number is NOT divisible by nine -- but what is the smallest amount you need to add to that number to make it divisible by 9? Hint

Add seven to make 45688

Add five to make 45686

Add three to make 45684

Add six to make 45687

NEXT>

Question 4 of 10

4. Another number is 468930. The even digits sum to 15, the odd digits to 15: 4+8+3 and 6+9+0 both equal 15.

Accordingly we can say with complete confidence that...? Hint

It is a perfect number

The number cannot be a square

The number is divisible by 11

The number has no odd factors

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Question 5 of 10

5. Let us take the number 15625. Which of these statements is NOT true? Hint

No numbers ending in seven are divisible by five

A number that ends in five must be prime

All numbers ending in five have only odd factors

Any number ending in five or zero is divisible by five

NEXT>

Question 6 of 10

6. Another example number is 48768. Which of these statements is true about any number ending in 8? Hint

It is precisely divisible by 8

It has no prime factors greater than 2

It cannot be divided by 7 with no remainder

It is not a perfect square

NEXT>

Question 7 of 10

7. If you add up three consecutive numbers the total will always be...?
So for example take 423 + 424 + 425; what generalization can be made? Hint

The sum will always be divisible by 6

The sum will always be even

The sum is divisible by 3

The sum will always be odd

8. We know 5 squared is 25. What rule can we apply to work out what the next higher square will be.
in other words to get from 5 squared to 6 squared will be, we take 25 and....?

Equally, to get from 11 squared equaling 121 to 12 squared, we simply.... Hint

The gap between squares will always be 15

Add 3 times the lower number, and take off 4

Add (5+6) to the lower square in one case, (11+12) in the second

The gap is always 11 between squares

NEXT>

Question 9 of 10

9. The fun fact about a right angle triangle is that the sum of the squares of the lengths of the two shorter sides = the sum of the square of the longer sides.
A right angle triangle with sides 3, 4, and 5 produces an example. 3 squared is 9, 4 squared is 16, and the sum of 25 equals 5 squared.
With a right angle triangle whose shortest side is 7, can you find the other two sides with whole numbers? Remember that 7 squared is 49.
Hint

16 and 17

24 and 25

9 and 10

32 and 33

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Question 10 of 10

10. Lets take a number with one unreadable digit:

7463#16

What should that missing digit be to give us a number divisible by 1, 2, 3, 4, 6, 8, 9 and 12?

Hint

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Quiz Answer Key and Fun Facts

1. Take a number, for example, 7846. Which of the following must be true about any number whose last digit is 6?

Answer: It cannot be prime

A prime number is divisible only by itself and 1.
Any number ending in an even digit, 0, 2, 4, 6, or 8 cannot be prime (unless it is 2 itself). It must be divisible by 2.
For the record the number 2 is prime, and is the only even prime.

2. Take a number, for example 46338. The digits add up to 24. What can be said about any number whose digits add to 24?

Answer: It is divisible by 3

The rule for determining if a number is divisible by three is to add the digits, and if the total is divisible by three (as 24 is) then the number itself is divisible by three.

3. Take the number 45681. The digits add to 24. As it stands the number is NOT divisible by nine -- but what is the smallest amount you need to add to that number to make it divisible by 9?

Answer: Add three to make 45684

The rule for being divisible by nine is similar to that for being divisible by three. The digits of any number that is divisible by 9 sum to a number divisible by 9. Hence 45684 sums to 27, and thus it is divisible by 9.

4. Another number is 468930. The even digits sum to 15, the odd digits to 15: 4+8+3 and 6+9+0 both equal 15. Accordingly we can say with complete confidence that...?

Answer: The number is divisible by 11

For all numbers divisible by 11 we can state accurately that the odd and even digits sum to the same number, or that the difference is divisible by 11.
For 3091 the odd digits sum to 12, the even ones to 1, and the number is therefore divisible by 11 (as a check: 11 x 281 is 3091).

5. Let us take the number 15625. Which of these statements is NOT true?

Answer: A number that ends in five must be prime

The only prime number that ends in five is five itself.

Five is the exception when it comes to looking at factors; all other numbers ending in five are divisible by five. 15625 is 25 cubed, and 125 squared.

An odd number can never have any even factors. This applies, of course, regardless of whether the odd number ends in 1, 3, 5, 7, or 9.

6. Another example number is 48768. Which of these statements is true about any number ending in 8?

Answer: It is not a perfect square

All perfect squares end in 0, 1, 4, 5, 6, and 9.
No perfect squares end in 2, 3, 7, or 8.

A simple way to see that is to work through the squares of all numbers from 1 to 9. None of them produce a number ending in eight. The last digit of any square therefore cannot be 8.

7. If you add up three consecutive numbers the total will always be...? So for example take 423 + 424 + 425; what generalization can be made?

Answer: The sum is divisible by 3

Think of this problem in general terms: if your smallest number is x, the second number will be x+1, the third x+2. Add the three together and you get 3x+3, so this number will always be divisible by three. Depending on whether the first number is even or odd the total may or may not be divisible by six, or even or odd.

8. We know 5 squared is 25. What rule can we apply to work out what the next higher square will be. in other words to get from 5 squared to 6 squared will be, we take 25 and....? Equally, to get from 11 squared equaling 121 to 12 squared, we simply....

Answer: Add (5+6) to the lower square in one case, (11+12) in the second

When you start with the square of 5 as 25, we get to 6 squared by adding 5+6 to 25 to reach 36. In the second case we add 11+12 or 23.

In algebraic terms if we know what x squared is, we add x, + x+1, or in this case 5+6 to 5 squared to get 6 squared. This is based on the square of (x+1) being x squared plus 2x+1, which can be restated to: x squared, plus x, plus x+1.

9. The fun fact about a right angle triangle is that the sum of the squares of the lengths of the two shorter sides = the sum of the square of the longer sides. A right angle triangle with sides 3, 4, and 5 produces an example. 3 squared is 9, 4 squared is 16, and the sum of 25 equals 5 squared. With a right angle triangle whose shortest side is 7, can you find the other two sides with whole numbers? Remember that 7 squared is 49.

Answer: 24 and 25

If you remember the last question, we demonstrated that the gap between the square of 24 and 25 is (24+25) which sums to 49. Seven squared is 49, so a triangle with sides 7, 24, 25 produces the desired answer.

10. Lets take a number with one unreadable digit: 7463#16 What should that missing digit be to give us a number divisible by 1, 2, 3, 4, 6, 8, 9 and 12?

Answer: 0

For a number to be divisible by all these factors the digits must sum to a number divisible by 9. Since the six digits we can see sum to 27, the missing digit must be 0 or 9. in either case we will achieve a number divisible by everything but 8. However to be divisible by 8 we must look at the last three digits only. 016 is divisible by 8, 916 is not. So the missing number is zero.

Source: Author garrybl

This quiz was reviewed by FunTrivia editor WesleyCrusher before going online.
Any errors found in FunTrivia content are routinely corrected through our feedback system.

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