Transcribed Image Text:443,692 and 11,092,785 for divisibility by 37.
19. Develop a test for divisibility by 37, based on the fact that 10 =1 (mod 37). Use this to check
15. Determine which base b repunits are divisible by factors of b – 1.
16. Determine which base b repunits are divisible by factors of b+ 1.
18. Show that every base 7 palindromic integer with an even number of digits is divisible by 8.
17. Show that every decimal palindromic integer with an even number of digits is divisible
5.1 Divisibility Tests
195
5.1 EXERCISES
1. Determine the highest power of 2 that divides each of the following positive integers.
a) 201,984
b) 1,423,408
c) 89,375,744
2. Determine the highest power of 5 that divides each of the following positive integers.
d) 41,578,912,246
a) 112,250
b) 4,860,625
c) 235,555,790
d) 48,126,953,125
3. Which of the following integers are divisible by 3? Of those that are, which are divisible
by 9?
a) 18,381
b) 65,412,351
c) 987,654,321
d) 78,918,239,735
4. Which of the following integers are divisible by 11?
a) 10,763,732
b) 1,086,320,015
c) 674,310,976,375
d) 8,924,310,064,537
5. Find the highest power of 2 that divides each of the following integers.
a) (101111110)2
b) (1010000011)2 c) (111000000)2
d) (1011011101)2
6. Determine which of the integers in Exercise 5 are divisible by 3.
7. Which of the following integers are divisible by 2?
a) (1210122)3
b) (211102101)3
c) (1112201112)3 d) (10122222011101)3
8. Which of the integers in Exercise 7 are divisible by 4?
9. Which of the following integers are divisible by 3, and which are divisible by 5?
a) (3EA235)16
b) (ABCDEF)16
c) (F117921173)16 d) (10AB987301F)16
10. Which of the integers in Exercise 9 are divisible by 17?
O A repunit is an integer with decimal expansion containing all 1s.
II. Determine which repunits are divisible by 3, and which are divisible by 9.
12. Determine which repunits are divisible by 11.
13. Determine which repunits are divisible by 1001. Which are divisible by 7? by 13?
14. Determine which repunits with fewer than 10 digits are prime.
A base b repunit is an integer with base b expansion containing all 1s.
15. Determine whích base b repunits are divisible by factors of b – 1.
A base b palindromic integer is an integer whose base b representation reads the same
forward and backward.
w that every decimal palindromic integer with an even number of digits is divisible
by 11.
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