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How many positive divisors do 8400 and 7560 have in common?

 Apr 24, 2018
 #1
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+1

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 10 | 12 | 14 | 15 | 16 | 20 | 21 | 24 | 25 | 28 | 30 | 35 | 40 | 42 | 48 | 50 | 56 | 60 | 70 | 75 | 80 | 84 | 100 | 105 | 112 | 120 | 140 | 150 | 168 | 175 | 200 | 210 | 240 | 280 | 300 | 336 | 350 | 400 | 420 | 525 | 560 | 600 | 700 | 840 | 1050 | 1200 | 1400 | 1680 | 2100 | 2800 | 4200 | 8400 (60 divisors of 8400)

 

 

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 12 | 14 | 15 | 18 | 20 | 21 | 24 | 27 | 28 | 30 | 35 | 36 | 40 | 42 | 45 | 54 | 56 | 60 | 63 | 70 | 72 | 84 | 90 | 105 | 108 | 120 | 126 | 135 | 140 | 168 | 180 | 189 | 210 | 216 | 252 | 270 | 280 | 315 | 360 | 378 | 420 | 504 | 540 | 630 | 756 | 840 | 945 | 1080 | 1260 | 1512 | 1890 | 2520 | 3780 | 7560 (64 divisors of 7560)

 

OK, young person! You just have to match them by counting them. You could also factor them first and calculate the divisors:

8400 = 2^4×3×5^2×7 (8 prime factors, 4 distinct)

7560 = 2^3×3^3×5×7 (8 prime factors, 4 distinct)

 Apr 24, 2018
edited by Guest  Apr 24, 2018
 #2
avatar+972 
+1

I think there is an easier way.

 

You want to find the GCF of the two numbers and see how many factors that number has.

 

GCF = 2 x 3 x 5 x 7 = 210

 

210 has

 

(1+1)^4 divisors.

 

16 will be your answer,

 

I hope this helps.

 

Gavin

 Apr 24, 2018
 #3
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Gavin: Your method doesn't seem to match, since there are about 30 divisors that the two numbers have in common!!.

 Apr 24, 2018
 #4
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Gavin's method appears to be sound, but I think he made a mistake in GCF or GCD:

 

GCD of {7,560, 8,400} =840

840 = 2^3 * 3 * 5 * 7

Then the number of divisors of 840 is =The product of the exponents of its factors + 1 for each factor:

(3+1)(1+1)(1+1)(1+1) =32 - Total divisors, which appears to agree with the actual count above.

 Apr 24, 2018

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