Download Blue – Prime Factorization DIVISIBILITY RULES 7

Unit 4 – Number Patterns and Fractions
Lesson 1 – Prime Factorization
Blue – Prime Factorization
DIVISIBILITY RULES
7
11
Take the last digit, double it, and subtract it from the rest of the number;
if the answer is divisible by 7 (including 0), then the number is also
Alternately add and subtract the digits from left to right. (You can think of the
first digit as being ‘added’ to zero.)
If the result (including 0) is divisible by 11, the number is also.
Example: to see whether 365167484 is divisible by 11, start by subtracting:
[0 +] 3 – 6 + 5 – 1 + 6 – 7 + 4 – 8 + 4 = 0; therefore 365167484 is divisible by 11.
Divisible
By 7?
by 11?
by 12?
781
1. ____
2. ____
3.
998
4. ____
5. ____
6.
1112
7. ____
8. ____
9.
364
10. ____
11. ____
12.
1482
13. ____
14. ____
15.
1212
16. ____
17. ____
18.
73080
19. ____
20. ____
21.
264
22. ____
23. ____
24.
4669
25. ____
26. ____
27.
____
____
____
____
____
____
____
____
____
28. Fill in the blanks (with one digit) so that the statements are true:
a. ____ 98 is divisible by 7
b. 78 ____ 2 is divisible by 9
c. 568 ____ is divisible by 12
29. Will the sum of any 2 prime numbers be prime? Justify your answer.
30. Twin primes are pairs of prime numbers that differ by two. The numbers 5 and 7 are
twin primes. Find all the twin primes that are less than 100.
31. Find the smallest 3-digit number that has 2, 3, 5 and another prime number as its
prime factors. List all other factors of the number.
32. Find the number that has exactly twelve factors (prime or composite) including 1 and
the number itself and is divisible by both 5 and 7.
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Unit 4 – Number Patterns and Fractions
Lesson 1 – Prime Factorization
33. Evaluate the expression n 2 + n + 11 for n = 0, 1, 2, and 3 to find four prime numbers.
What is the smallest whole number that you could substitute in for n which does not
result in a prime number?
34. Find a number between 40 and 50 that is composite and whose prime factors add up
to 15.
35. Find a number between 41 and 50 whose prime factors add up to 25.
36. There are four consecutive prime numbers written in ascending order. The product
for the first three is 7429 and the last three is 12673. Find the numbers.
37. Three consecutive positive integers are such that the product of the 1st and 3rd is
288. Find the middle number.
38. Which number is always a factor of a two digit number plus those same 2 digits
reversed (for example 45 + 54).
39. Assembly Seating
Ben was helping the school custodian set up chairs in the gym for an assembly. They
needed to seat 180 students and put the same number of chairs in each row.
After hearing the custodian's plan and doing some quick mental math, Ben suggested
a different arrangement for the same number of seats. He explained that, by putting
5 more chairs in each row, they could have 3 fewer rows, and students in the back
row would be able to see better.
How many chairs were in each row of Ben's plan?
Explain how you solved the problem.
40. Four Playful Mathematicians
Four playful mathematicians were eating lunch together at an outdoor cafe. They noticed a
diamond-encrusted Rolex had been left behind at the table next to them, so they quickly
looked around for the person who had vacated the table. They saw the owner of the watch
driving away. "Quick, get the license plate!" urged one of the mathematicians, seeing that
it was only a four-digit number.
When the car had disappeared around the corner, they put together these characteristics
they'd noticed about the number:
1. The first said, "One of the digits is neither prime nor composite."
2. The second contributed, "The digit in the hundreds place is three times the
digit in the thousands place."
3. The third added, "The digit in the ones place is four times greater than the
digit in the tens place."
4. The fourth piped up with the last bit of information. "The smallest prime
number is in the thousands place. We have enough information to help us locate
the owner of the watch."
Question: What number did the four playful mathematicians see on the license
plate?
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Unit 4 – Number Patterns and Fractions
Lesson 1 – Prime Factorization
Solutions
1. No
2. Yes
3. No
4. No
5. No
6. No
7. No
8. No
9. No
10. Yes
11. No
12. No
13. No
14. No
15. No
16. No
17. No
18. Yes
19. Yes
20. No
21. Yes
22. No
23. Yes
24. Yes
25. Yes
26. No
27. No
28. a. 7
b. 1
c. 8
29. No; Examples may vary (3 + 5 = 8. 8 is not prime.)
30. 3:5 / 5:7/ 11:13 / 17:19 / 29:31 / 41:43 / 59:61 / 71:73 / 89:91
31. 210 → 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210
32. 140
33. 10
34. 44
35. 46
36. 17, 19, 23, 29
37. 17
38. 11
39. First, I listed all of the factors and factor pairs of the total number of seats (180)
because the problem stated that they put the same number of seats in each row.
These are the factor pairs I got:
1,
180
2,
90
3,
60
4,
45
5,
36
6,
30
9,
20
10,
18
12,
15
I then looked for a difference of three with a difference of
5 between two factor pairs because the problem said that Ben's plan
enabled them to take off three rows from the custodian's plan, but
add five seats to each row. I found a difference of three and five
between the 7th and 9th factor pair (9,20 and 12,15). That is how I
got my answer of 20 chairs.
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Unit 4 – Number Patterns and Fractions
Lesson 1 – Prime Factorization
40. First, I figured out that the thousands place is a 2. I found this out because the clue
says "The smallest prime number is in the thousands place." and two is the smallest
prime number.
Second, I figured out that the hundreds place is a 6 because the clue says "The digit
in the hundreds place is three times the digit in the thousands place." and 2x3=6
Next, I figured out that the tens place is a 1. I found this out because the clue says
"One of the digits is neither prime nor composite." and a composite number is a
number bigger than one that has more than two factors. A prime number is a number
that's only factors are one and itself. This could also be 0, but if it was, then there
would be a 0 in the ones place also. Therefore, the answer is 1.
Finally, I figured out that the ones place is 4. I found this out because the clue says
"The digit in the ones place is four times the digit in the tens place." The one has to
be in the tens place to be able to find out which number is in the ones place.
Therefore, 1 x 4 = 4.
Therefore, the answer is 2614.
4
Unit 4 – Number Patterns and Fractions
Lesson 1 – Prime Factorization
Bibliography Information
Teachers attempted to cite the sources for the problems included in this problem set. In some cases,
sources may not have been known.
Problems
Bibliography Information
39-40
The Math Forum @ Drexel
(http://mathforum.org/)
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