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Quiz about Divisibility Rules Divisibility Rules
Quiz about Divisibility Rules Divisibility Rules

Divisibility Rules? Divisibility Rules! Quiz

Is 21 divisible by 7? That's easy. But is 1696968 divisible by 7? Looks like a tougher nut to crack. But it isn't... This quiz will introduce you to a few basic divisibility rules. My approach here is practical, rather than theoretical.

A multiple-choice quiz by gentlegiant17. Estimated time: 5 mins.
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Author
gentlegiant17
Time
5 mins
Type
Multiple Choice
Quiz #
253,492
Updated
Jul 23 22
# Qns
10
Difficulty
Tough
Avg Score
6 / 10
Plays
879
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Question 1 of 10
1. How can one tell if an integer is divisible by 6? Hint


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Question 2 of 10
2. Which of the following is a rule for divisibility by 4? Hint


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Question 3 of 10
3. 999999 is evenly divisible by 7.


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Question 4 of 10
4. Which of the following integers is evenly divisible by 11? Hint


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Question 5 of 10
5. Do you know these ultra-superstitious people? I knew someone who refused to have anything to do with the number 13, or any of its multiples. He once rejected 3 phone numbers offered by his cellphone operator on this very ground. Which number was he willing to accept eventually? Hint


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Question 6 of 10
6. Fill in the blank digit in order for the resultant number to be divisible by 45: 5_1_5_1_5 (the same number goes in all the blanks.)

Answer: (one digit, think prime factorization)

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Question 7 of 10
7. Which of the following statements about the integer 9699690 is correct? Hint


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Question 8 of 10
8. How many of the first 1000 prime integers have divisibility rules? Hint


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Question 9 of 10
9. Let's move on from the specific decimal realm to generic bases. For this question, our base is B.

Complete the following rule: "If the sum of digits of an integer is divisible by ___ then the integer is divisible by ___."

Answer: (Same expression in both blanks - a simple function of B)

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Question 10 of 10
10. If k is a prime factor of the base B, then a number is divisible by k^n if and only if the last n digits of the number are divisible by k^n.



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Quiz Answer Key and Fun Facts
1. How can one tell if an integer is divisible by 6?

Answer: The last digit is even and the sum of digits is divisible by 3

 An easy one to start with, building on the divisibility rules for 2 (last digit is even) and 3 (sum of digits is divisible by 3). Divisibility by 6 means divisibility by both 2 and 3 (prime factors of 6), so both rules have to be applied.
2. Which of the following is a rule for divisibility by 4?

Answer: All three are valid

 All three rules apply, for example:

28 - the tens digit is even and the last digit is 0, 4 or 8
32 - the tens digit is odd and the last digit is 2 or 6
128,132 - the number formed by the last two digits is divisible by 4
3. 999999 is evenly divisible by 7.

Answer: True

999999/7=142857.

There are several divisibility rules for 7. Let's explore one of them:

Take the last digit, multiply by 2 and subtract from the remaining number: 99999-2*9=99981. Go on: 9998-2*1=9996. Go on: 999-2*6=987. Go on: 98-2*7=84. Go on: 8-2*4=0.

If you have reached 0,7 or -7 at the end of the chain - the number in question is divisible by 7.
4. Which of the following integers is evenly divisible by 11?

Answer: 855503

 The rule I like the most for 11 is to create the two sums of alternating digits, subtract them and check if we get to 0 or to 11.

855503: (8+5+0)-(5+5+3)=0 divisible by 11
855513: (8+5+1)-(5+5+3)=1 not divisible by 11
855523: (8+5+2)-(5+5+3)=2 not divisible by 11
855533: (8+5+3)-(5+5+3)=3 not divisible by 11
5. Do you know these ultra-superstitious people? I knew someone who refused to have anything to do with the number 13, or any of its multiples. He once rejected 3 phone numbers offered by his cellphone operator on this very ground. Which number was he willing to accept eventually?

Answer: 4826808

 Only 4826808 is not divisible by 13 (actually it is 13^6-1).

The are several divisibility rules for 13. Let's explore one of them to learn about 6815705:

Take the last digit, multiple by 9 and subtract from the remaining number: 681570-9*5=681525. Go on: 68152-9*5=68107. Go on: 6810-9*7=6747. Go on: 674-9*7=611. Go on: 61-9*1=52. Go on: 5-9*2=-13.

If you have reached 0,13 or -13 at the end of the chain - the number in question is divisible by 13.
6. Fill in the blank digit in order for the resultant number to be divisible by 45: 5_1_5_1_5 (the same number goes in all the blanks.)

Answer: 7

In order for an integer to be divisible by 45, it has to be divisible by the prime factors of 15: 3,3 and 5 which cuts down to 9 and 5. In terms of divisibility rules, this comes down to the demand that both the sum of digits would be divisible by 9, and that the last digit would be either 5 or 0 (which is already given).
In this example only 7 contributes to the fulfillment of the divisibility by 9 requirement.
7. Which of the following statements about the integer 9699690 is correct?

Answer: It is divisible by all prime numbers up to 19 (inclusive)

 9699690=1*2*3*5*7*11*13*17*19

Divisibility by 17 is determined by the same method shown for 7 - only instead of 2, use 5 as the multiplier for the last digit. For 19, use 17. For 23, use 16. For 29, use 26.
8. How many of the first 1000 prime integers have divisibility rules?

Answer: All of them

 Not a big surprise by now, but all of them do.

For the gory details, take a look in http://arxiv.org/ftp/math/papers/0001/0001012.pdf
9. Let's move on from the specific decimal realm to generic bases. For this question, our base is B. Complete the following rule: "If the sum of digits of an integer is divisible by ___ then the integer is divisible by ___."

Answer: B-1

In base 10 (aka decimal), the divisibility rule for 9 is well known -if the sum of digits of an integer is divisible by 9 (10-1) then the integer is divisible by 9 (10-1).

This is a private case of the more general rule outlined in the question.

Example in hexadecimal (base 16): 0xE1 (225 decimal) is divisible by 0xF (16-1=15 decimal) since 0xE+0x1=0xF is obviously divisible by 0xF.
10. If k is a prime factor of the base B, then a number is divisible by k^n if and only if the last n digits of the number are divisible by k^n.

Answer: True

Lovely rule.

In base 10, it will give you divisibility rules for 2,5 and all of their powers.
Source: Author gentlegiant17

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