Exercise 1

Determine all of the multiples of 17 that exist between 800 and 860.

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Exercise 2

For the following numbers: 179, 311, 848, 3,566, 7,287, indicate which are prime and which composite numbers.

Exercise 3

Determine, using a table, all the prime numbers between 400 and 450.

Exercise 4

Factor the following numbers:

  1. 216
  2.  360
  3.  432

Exercise 5

Factor 342 and determine its number of divisors.

Exercise 6

Factor the following numbers:

  1. 2,250
  2. 428 and 376
  3. 3,500
  4. 2,520

Exercise 7

Calculate the greatest common divisor (GCD) and the lowest common multiple (LCM) of the following numbers:

  1. 148 and 156
  2. 600 and 1,000

Exercise 8

Calculate the greatest common divisor (GCD) and the lowest common multiple (LCM) of the following numbers:

  1. 72, 108 and 600
  2. 1,048, 786 and 3,930
  3. 3,120, 6,200 and 1,864

Exercise 9

Determine, using the Euclidean algorithm, the greatest common divisor (GCD) of the following:

  1. 72 and 16
  2. 656 and 848
  3. 1,278 and 842

 

Solution of exercise 1

Determine all of the multiples of 17 that exist between 800 and 860.

816, 833, 850

 

Solution of exercise 2

For the following numbers: 179, 311, 848, 3,566, 7,287, indicate which are prime and which composite numbers.

Prime numbers: 179 and 311 because each has only two multiples, 1 and the number itself.

Composite numbers : 848, 3,566 and 7,287. These numbers have more than two multiples, so we have categorized them as composite numbers.

 

Solution of exercise 3

Determine, using a table, all the prime numbers between 400 and 450.

401 409
419
421
431 433 439
443 449

 

Solution of exercise 4

Factor the following numbers:

  1. 216
2216
2108
254
327
39
33
1

216 = 2 ^ 3 \cdot 3 ^ 3

      2. 360

2360
2180
290
345
315
55
1

 

360 = 2 ^ 3 \cdot 3 ^ 2 \cdot 5

    3. 432

2432
2216
2108
254
327
39
33
31

432 = 2^ 4 \cdot  3 ^ 3

 

Solution of exercise 5

Factor 342 and determine its number of divisors.

342 = 2 \cdot 3 ^ 2 \cdot 19

n = (1 + 1) \cdot (2 + 1) \cdot (1 + 1) = 12

 

Solution of exercise 6

Factor the following numbers:

  1. 2,250

22250
31125
3375
5125
321
77
1

2250 = 2 \cdot 3 ^ 2 \cdot 5 ^ 3

2.   3,500

23500
21750
5875
5175
535
77
1

3500 = 2 ^ 2 \cdot 5 ^ 3 \cdot 7

       3.  2,520

22520
21260
2630
3315
3105
535
77
1

2520 = 2 ^3 \cdot 3 ^2 \cdot 5 \cdot 7

 

Solution of exercise 7

Calculate the greatest common divisor (GCD) and the lowest common multiple (LCM) of the following numbers:

 

1. 428 and 376

428 = 2 ^ 2 \cdot 107

376 = 2 ^3 \cdot 47

G.C.D. (428 , 376) = 2 ^2 = 4

L.C.M. (428 , 376) = 2 ^3 \cdot 107 \cdot 47 = 40232

 

2.  148 and 156

148 = 2 ^ 2 \cdot 37

156 = 2 ^2 \cdot 3 \cdot 13

GCD (148 , 156) = 2 ^2 = 4

LCM (148 , 156) = 2 ^2 \cdot 3 \cdot 37 \cdot 13 = 5772

 

3.  600 and 1,000

600 = 2 ^3 \cdot 3 \cdot 5 ^2

1000 = 2 ^3 \cdot 5 ^3

GCD (600 , 1000) = 2 ^3 \cdot 5 ^2 = 200

LCM (600 , 1000) = 2 ^3 \cdot 3 \cdot 5 ^3 = 3000

 

Solution of exercise 8

Calculate the greatest common divisor (GCD) and the lowest common multiple (LCM) of the following numbers:

           1. 72, 108 and 60.

72 = 2 ^3 \cdot 3 ^2

108 = 2 ^2 \cdot 3 ^3

60 = 2 ^2 \cdot 3 \cdot 5

GCD (72 , 108 , 60) = 2 ^2 \cdot 3

LCM (72 , 108 , 60) = 2 ^3 \cdot 3 ^3 \cdot 5 = 2160

 

       2.  1,048, 7,86 and 3,930
21048
2524
2262
131131
1
1048 = 2 ^3 \cdot 131
2786
3393
131131
1
786 = 2 \cdot 3 \cdot 131
23930
31965
5655
131131
1
3930 = 2 \cdot 3 \cdot 5 \cdot 131GCD (1048 , 786 , 3930) = 2 \cdot 131 = 262LCM (1048 , 786 , 3930) = 2 ^3 \cdot 3 \cdot 5 \cdot 131 = 15720

 

       3.   3,120, 6,200 and 1,864

23120
21560
2780
2390
5195
339
1313
1

3210 = 2 ^4 \cdot 3 \cdot 5 \cdot 13

26200
23100
21550
5775
5155
3131
1

6200 = 2 ^3 \cdot 5 ^2 \cdot 31

21864
2932
2416
2208
2104
252
226
1313
1

1864 = 2 ^3 \cdot 233

GCD (3210 , 6200 , 1864) = 2 ^3 = 8

LCM (3210 , 6200, 1864) = 2 ^4 \cdot 3 \cdot 5 ^2 \cdot 31 \cdot 233 = 112678800

 

Solution of exercise 9

Determine, using the Euclidean algorithm, the greatest common divisor (GCD) of:

        1. 72 and 16

\begin{array}{ccccccccccc} & & &  4&&&&\\ \cline{3-7} \multicolumn{2}{r}{16 \surd} && 72 && &&&&\\ & &&64& & & & \\ \cline{3-7} & && 8& & \\ \end{array}

 

\begin{array}{ccccccccccc} & & &  2&&&&\\ \cline{3-7} \multicolumn{2}{r}{8 \surd} && 16 && &&&&\\ & &&16& & & & \\ \cline{3-7} & & &0& & \\ \end{array}

 

GCD (72, 16) = 8

 

     2.  656 and 848

\begin{array}{ccccccccccc} & & &  1&&&&\\ \cline{3-7} \multicolumn{2}{r}{656 \surd} && 848&& &&&&\\ & &&656& & & & \\ \cline{3-7} & && 192& & \\ \end{array}

 

\begin{array}{ccccccccccc} & & &  3&&&&\\ \cline{3-7} \multicolumn{2}{r}{192 \surd} && 656&& &&&&\\ & &&576& & & & \\ \cline{3-7} & && 80& & \\ \end{array}

 

\begin{array}{ccccccccccc} & & &  2&&&&\\ \cline{3-7} \multicolumn{2}{r}{80\surd} && 192&& &&&&\\ & &&160& & & & \\ \cline{3-7} & && 32& & \\ \end{array}

 

\begin{array}{ccccccccccc} & & &  2&&&&\\ \cline{3-7} \multicolumn{2}{r}{32\surd} && 80&& &&&&\\ & &&64& & & & \\ \cline{3-7} & && 16& & \\ \end{array}

 

\begin{array}{ccccccccccc} & & &  2&&&&\\ \cline{3-7} \multicolumn{2}{r}{26\surd} && 32&& &&&&\\ & &&32& & & & \\ \cline{3-7} & && 0& & \\ \end{array}

 

GCD (656, 848) = 16

          3.  17,28 and 842

\begin{array}{ccccccccccc} & & &  2&&&&\\ \cline{3-7} \multicolumn{2}{r}{842\surd} && 1728&& &&&&\\ & &&1684& & & & \\ \cline{3-7} & && 44& & \\ \end{array}

 

\begin{array}{ccccccccccc} & & &  19&&&&\\ \cline{3-7} \multicolumn{2}{r}{44\surd} && 842&& &&&&\\ & &&836& & & & \\ \cline{3-7} & && 6& & \\ \end{array}

 

 

\begin{array}{ccccccccccc} & & &  7&&&&\\ \cline{3-7} \multicolumn{2}{r}{6\surd} && 44&& &&&&\\ & &&42& & & & \\ \cline{3-7} & && 2& & \\ \end{array}

 

 

\begin{array}{ccccccccccc} & & &  3&&&&\\ \cline{3-7} \multicolumn{2}{r}{2\surd} && 6&& &&&&\\ & &&6& & & & \\ \cline{3-7} & && 0& & \\ \end{array}

GCD (1,278, 842) = 2

 

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.