A multiple-choice quiz
by Matthew_07.
Estimated time: 4 mins.

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Quiz Answer Key and Fun Facts

Answer:
**2**

In the fields of real analysis and discrete mathematics, a prime number is defined as a number that is greater than 1 and has no positive divisors other than 1 and the number itself. So, by this definition, 1 is not a prime number. The divisors of 2 are 1 and 2. So 2 fits the description of a prime number.

The next number, 3, also fits the criteria and hence, is also a prime number. The divisors of the next number, 4, are 1, 2, and 4. Using the definition given, 4 is not a prime number. It is a composite number from the fact that 4 = 2 x 2.

In the fields of real analysis and discrete mathematics, a prime number is defined as a number that is greater than 1 and has no positive divisors other than 1 and the number itself. So, by this definition, 1 is not a prime number. The divisors of 2 are 1 and 2. So 2 fits the description of a prime number.

The next number, 3, also fits the criteria and hence, is also a prime number. The divisors of the next number, 4, are 1, 2, and 4. Using the definition given, 4 is not a prime number. It is a composite number from the fact that 4 = 2 x 2.

Answer:
**0**

Zero is neither positive nor negative. Zero is actually an even number. In mathematics, an even number is defined as a number that can be written in the form of 2k, where k is an integer. Notice that 0 = 2 x 0. The number zero itself is also an integer.

The numbers 2 and 4 are positive even numbers. On the other hand, -2 and -4 are negative even numbers.

Zero is neither positive nor negative. Zero is actually an even number. In mathematics, an even number is defined as a number that can be written in the form of 2k, where k is an integer. Notice that 0 = 2 x 0. The number zero itself is also an integer.

The numbers 2 and 4 are positive even numbers. On the other hand, -2 and -4 are negative even numbers.

Answer:
**7**

Notice that 1 + 2 + 3 + 4 + 5 + 6 = 21. 21/3 = 7. So even if you have no idea on the arrangement of the pips or values on a die, a clever and educated guess is 7.

Notice that 1 + 2 + 3 + 4 + 5 + 6 = 21. 21/3 = 7. So even if you have no idea on the arrangement of the pips or values on a die, a clever and educated guess is 7.

Answer:
**999999**

A few more interesting properties of the number 142857:

142 + 857 = 999

14 + 28 + 57 = 99

142857 x 142857 = 20408122449

20408 + 122449 = 142857

A few more interesting properties of the number 142857:

142 + 857 = 999

14 + 28 + 57 = 99

142857 x 142857 = 20408122449

20408 + 122449 = 142857

Answer:
**5**

Notice that 12, 22, 32, 42, 52, ... are all even numbers and hence, divisible by 2. In addition, 15, 25, 35, 45, 55, ... are all multiples of 5, so they are all composite numbers.

Notice that 12, 22, 32, 42, 52, ... are all even numbers and hence, divisible by 2. In addition, 15, 25, 35, 45, 55, ... are all multiples of 5, so they are all composite numbers.

Answer:
**6**

1 week = 7 days

1 day = 24 hours

1 hour = 60 minutes

1 minute = 60 seconds

Using the four formulas above, and the fact that 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1, we can convert the units easily.

(10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / [60 x 60 x 24 x 7]

= (10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / [(2 x 5 x 6) x (6 x 10) x (3 x 8) x 7]

(cancelling out the common terms)

= ( 9 x 4 ) / ( 6 )

= 36 / 6

= 6 weeks

It is interesting to note that a seemingly large number, 10!, can be reduced to an elegant integer, 6, using some common conversion units.

1 week = 7 days

1 day = 24 hours

1 hour = 60 minutes

1 minute = 60 seconds

Using the four formulas above, and the fact that 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1, we can convert the units easily.

(10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / [60 x 60 x 24 x 7]

= (10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / [(2 x 5 x 6) x (6 x 10) x (3 x 8) x 7]

(cancelling out the common terms)

= ( 9 x 4 ) / ( 6 )

= 36 / 6

= 6 weeks

It is interesting to note that a seemingly large number, 10!, can be reduced to an elegant integer, 6, using some common conversion units.

Answer:
**900**

The formula (a + b)^2 = a^2 + b^2 is a very common misconception. The correct formula should be (a + b)^2 = a^2 + b^2 + 2ab.

Let a = 11 and b = 19. (11 + 19)^2 = 11^2 + 19^2 + 2x11x19 = 121 + 361 + 418 = 900. We can double check the answer: (11 + 19)^2 = 30^2 = 900.

The formula (a + b)^2 = a^2 + b^2 is a very common misconception. The correct formula should be (a + b)^2 = a^2 + b^2 + 2ab.

Let a = 11 and b = 19. (11 + 19)^2 = 11^2 + 19^2 + 2x11x19 = 121 + 361 + 418 = 900. We can double check the answer: (11 + 19)^2 = 30^2 = 900.

Answer:
**24**

A systematic way to determine the least common multiple (LCM) of any two integers is to list down all the multiples for each of the two numbers.

Multiples of 6: 6, 12, 18, 24, 30, 36...

Multiples of 8: 8, 16, 24, 32, 40, 48...

Observe that the number 24 appears in both the lists. So 24 is the LCM of 6 and 8.

Another more efficient way to find the LCM of two integers is by prime factorization.

Notice that 6 = 2 x 3 and 8 = 2 x 2 x 2 = 2^3. We first determine the highest power of all the prime numbers. Then, we multiply them to get the value of the LCM. In this case, we have 2^3 x 3 = 24.

Simply multiplying the given two numbers together will not necessarily give the LCM of the two numbers. 6 x 8 = 48. 48 is a common multiple of 6 and 8 but it is not the least common multiple.

However, in a more specific case where both the given numbers are prime numbers, we can simply multiply the two numbers together to get the LCM. For example, the LCM of 5 and 7 is simply 5 x 7 = 35.

A systematic way to determine the least common multiple (LCM) of any two integers is to list down all the multiples for each of the two numbers.

Multiples of 6: 6, 12, 18, 24, 30, 36...

Multiples of 8: 8, 16, 24, 32, 40, 48...

Observe that the number 24 appears in both the lists. So 24 is the LCM of 6 and 8.

Another more efficient way to find the LCM of two integers is by prime factorization.

Notice that 6 = 2 x 3 and 8 = 2 x 2 x 2 = 2^3. We first determine the highest power of all the prime numbers. Then, we multiply them to get the value of the LCM. In this case, we have 2^3 x 3 = 24.

Simply multiplying the given two numbers together will not necessarily give the LCM of the two numbers. 6 x 8 = 48. 48 is a common multiple of 6 and 8 but it is not the least common multiple.

However, in a more specific case where both the given numbers are prime numbers, we can simply multiply the two numbers together to get the LCM. For example, the LCM of 5 and 7 is simply 5 x 7 = 35.

Answer:
**978**

Here's a simple example to show why the rule "if the sum of all the individual digits of a number is divisible by 3, then the number itself is also divisible by 3" is true for any number.

We will use a 3-digit number to show how this works, but the result can be generalized to any other numbers.

Let's try with the number 432. We can decompose the number into 4 x 100 + 3 x 10 + 2 x 1. We can then rewrite it as 4 x (99 + 1) + 3 x (9 + 1) + 2 = (4 x 99) + (3 x 9) + (4 + 3 + 2). Notice that 99 and 9 are multiples of 3, which implies that 4 x 99 and 3 x 9 are both divisible by 3. This leaves us with the operation 4 + 3 + 2. If the sum is also divisible by 3, then the number 432 is also divisible by 3. 4 + 3 + 2 = 9 is divisible by 3. So we conclude that the number 432 is divisible by 3.

A more rigorous mathematical proof involves the concept of modulo arithmetic. Any number in the form of 10^n is equivalent to 1 modulo 3, where n is any positive integer. In layman's terms, for n = 1, when you divide 10 by 3, you will obtain a remainder of 1. In fact, you will get the same remainder when you perform the division operation on 10^2 = 100, 10^3 = 1000, ... , 10^n. This property is based on the fact that 10^n = 99...99 + 1, and the term 99...99 is divisible by 3.

To find the greatest 3-digit that ends in 8 that is divisible by 3, we know that the number takes the form of 9_8. The number 998 might seem to be answer but 9 + 9 + 8 = 26 implies that 998 is not divisible by 3. We proceed to the next possible value, which is 988, but again, 9 + 8 + 8 = 25 is not divisible by 3. We then try the number 978. The sum of its digits, 9 + 7 + 8 = 24 is a multiple of 3. So the number 978 is the number that fits the given criteria.

Here's a simple example to show why the rule "if the sum of all the individual digits of a number is divisible by 3, then the number itself is also divisible by 3" is true for any number.

We will use a 3-digit number to show how this works, but the result can be generalized to any other numbers.

Let's try with the number 432. We can decompose the number into 4 x 100 + 3 x 10 + 2 x 1. We can then rewrite it as 4 x (99 + 1) + 3 x (9 + 1) + 2 = (4 x 99) + (3 x 9) + (4 + 3 + 2). Notice that 99 and 9 are multiples of 3, which implies that 4 x 99 and 3 x 9 are both divisible by 3. This leaves us with the operation 4 + 3 + 2. If the sum is also divisible by 3, then the number 432 is also divisible by 3. 4 + 3 + 2 = 9 is divisible by 3. So we conclude that the number 432 is divisible by 3.

A more rigorous mathematical proof involves the concept of modulo arithmetic. Any number in the form of 10^n is equivalent to 1 modulo 3, where n is any positive integer. In layman's terms, for n = 1, when you divide 10 by 3, you will obtain a remainder of 1. In fact, you will get the same remainder when you perform the division operation on 10^2 = 100, 10^3 = 1000, ... , 10^n. This property is based on the fact that 10^n = 99...99 + 1, and the term 99...99 is divisible by 3.

To find the greatest 3-digit that ends in 8 that is divisible by 3, we know that the number takes the form of 9_8. The number 998 might seem to be answer but 9 + 9 + 8 = 26 implies that 998 is not divisible by 3. We proceed to the next possible value, which is 988, but again, 9 + 8 + 8 = 25 is not divisible by 3. We then try the number 978. The sum of its digits, 9 + 7 + 8 = 24 is a multiple of 3. So the number 978 is the number that fits the given criteria.

Answer:
**41 x 32**

From the given four values, namely 1, 2, 3, and 4, we know that in order to make the product as big as possible, the two numbers should be in the form of 4_ and 3_. Now we need to consider where to put the remaining two numbers, 1 and 2. The two possible combinations are 41 x 32 and 42 x 31. Observe that 41 x 32 = 1312 > 42 x 31 = 1302.

A more challenging question requires you to determine the largest product that you can obtain by forming two values from the given nine numbers, 1 until 9. Using the concept of rearrangement inequality, the answer is 9642 × 87531 = 843973902.

From the given four values, namely 1, 2, 3, and 4, we know that in order to make the product as big as possible, the two numbers should be in the form of 4_ and 3_. Now we need to consider where to put the remaining two numbers, 1 and 2. The two possible combinations are 41 x 32 and 42 x 31. Observe that 41 x 32 = 1312 > 42 x 31 = 1302.

A more challenging question requires you to determine the largest product that you can obtain by forming two values from the given nine numbers, 1 until 9. Using the concept of rearrangement inequality, the answer is 9642 × 87531 = 843973902.

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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