Answer to Question #348679 in Discrete Mathematics for Ttypre

Question

A. Determine if 1781 is divisible by 3, 6, 7, 8, and 9. (5 items x 2 points)

B. Determine if each of the following numbers is a prime or composite.

6. 828

7. 1666

8. 1781

9. 1125

10. 107

C. Find the greatest common divisor of each of the following pairs of integers.

11. 60 and 100

12. 45 and 33

13. 34 and 58

14. 77 and 128

15. 98 and 273

D. Find the least common multiple of each of the following pairs of integers.

16. 72 and 108

17. 175 and 245

18. 150 and 70

19. 32 and 27

20. 540 and 504

Answer 1

A. The number 1781 is odd. Then it is not divisible by 6 and 8.

Determine if 1781 is divisible by 3, 6, 7, 8, and 9.

The sum of digits of the number is $1+7+8+1=17.$ 17 is not divisible by 3. Then 1781 is not divisible by 3, 6, and 9.

$$1781=200(7)+50(7)+4(7)+3$$

1781 is not divisible by 7.

1781 is not divisible by 3, 6, 7, 8, and 9.

B.

6. 828 is even => is divisible by 2 => is a composite.

7. 1666 is even => is divisible by 2 => is a composite.

8. 1781=13(137) => is divisible by 13 => is a composite.

9. 1125 is divisible by 5 => is a composite.

10. 107 is the 28th prime number.

$$10^2<107<11^2$$

107 is not divisible by 2, 3, 4, 5, 6, 7, 8,9, 10, and 11=> is a prime.

C. Find the greatest common divisor of each of the following pairs of integers.

11.

$$60=2(2)(3)(5)$$

$$100=2(2)(5)(5)$$

The greatest common divisor $2(2)(5)=20$

12.

$$45=3(3)(5)$$

$$33=3(11)$$

The greatest common divisor $3$

13.

$$34=2(17)$$

$$58=2(29)$$

The greatest common divisor $2$

14.

$$77=7(11)$$

$$128=2(2)(2)(2)(2)(2)(2)$$

The greatest common divisor $1$

15.

$$98=2(7)(7)$$

$$273=3(7)(13)$$

The greatest common divisor $7$

D. Find the least common multiple of each of the following pairs of integers.

16.

$$72=2(2)(2)(3)(3)$$

$$108=2(2)(3)(3)(3)$$

The least common multiple $2(2)(2)(3)(3)(3)=216$

17.

$$175=5(5)(7)$$

$$245=5(7)(7)$$

The least common multiple $5(5)(7)(7)=1225$

18.

$$150=2(3)(5)(5)$$

$$70=2(5)(7)$$

The least common multiple $2(3)(5)(5)(7)=1050$

19.

$$32=2(2)(2)(2)(2)$$

$$27=3(3)(3)$$

The least common multiple $2(2)(2)(2)(2)(3)(3)(3)=864$

20.

$$540=2(2)(3)(3)(3)(5)$$

$$504=2(2)(2)(3)(3)(7)$$

The least common multiple $2(2)(2)(3)(3)(3)(5)(7)=7560$