Which of the following is divisible by the first six positive integer multiples of 19?
(i) 52440
(ii) 114600
(iii) 19038	Select:
A) None
B) i only
C) iii only
D) ii and iii only
E) i and ii only
A positive integer x is divisible by 19, 2*19,...,6*19 if and only if it is divisible by their least common multiple, which in this case is 3*4*5*19. In particular, x is divisible by 10, so iii is not divisible by the first six positive integer multiples of 19.   3*4*5*19=60*(20-1)= 
<gadget id="calculator">60*(20-1)</gadget>
<output>1_140</output>
1140. 1140*100=114000, so clearly (ii) is not divisible by the first six positive integer multiples of 19.   It remains to verify (i). 52440-40*1140=52440-45600= 
<gadget id="calculator">52440-45600</gadget>
<output>6_840</output>
6840=6*1140, so (i) is divisible by 1140.   Answer i only.
<result>B</result>
Which of the following is divisible by the first six positive integer multiples of 19?
(i) 52440
(ii) 114600
(iii) 19038