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To have a gcf of 18 they must have one factor of 18 in common and no other factor, ie the second factor of the numbers must be relatively prime.
The multiples of 18 between 200 and 300 are:
12 x 18 = 216
13 x 18 = 234
14 x 18 = 252
15 x 18 = 270
16 x 18 = 288
12 is relatively prime to 13
13 is relatively prime to 14, 15 & 16
14 is relatively prime to 15
15 is relatively prime to 16
(Pairs of numbers are being considered, so as 12 is relatively prime to 13, 13 is also relatively prime to 12 and thus is only listed when the numbers increase, ie under 12 and not under 13.)
Thus the pairs:
216 & 234
234 & 252
234 & 270
234 & 288
252 & 270
270 & 288
all have a gcf of 18.
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Required numbers from 200 and 300 are 207,217,227,237,247,257,267,270,271,272,273,274,275,276,278,279,287,297 these are the numbers therefore, the answer is 18
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