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16 Sep 2014, 01:30
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
60% (02:17) correct
40%
(02:15)
wrong
based on 317
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Not Attempted Yet
If the product of a positive integer \(x\) and 377,910 is divisible by 3,300, what is the least possible value of \(x\)?
A. \(10\)
B. \(11\)
C. \(55\)
D. \(110\)
E. \(330\)
A. \(10\)
B. \(11\)
C. \(55\)
D. \(110\)
E. \(330\)
16 Sep 2014, 01:31
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Official Solution:
If the product of a positive integer \(x\) and 377,910 is divisible by 3,300, what is the least possible value of \(x\)?
A. \(10\)
B. \(11\)
C. \(55\)
D. \(110\)
E. \(330\)
Given: \(\frac{377,910*x}{3,300}=integer\).
Factorize the divisor: \(3,300=2^2*3*5^2*11\).
Check 377,910 for divisibility by \(2^2\): 377,910 IS divisible by 2 and NOT divisible by \(2^2 = 4\) (since its last two digits, 10, is not divisible by 4). Thus \(x\) must have 2 as its factor (377,910 is divisible only by 2, so in order for \(377,910*x\) to be divisible by \(2^2\), \(x\) must have 2 as its factor);
Check 377,910 for divisibility by 3: 3 + 7 + 7 + 9 + 1 + 0 = 27, thus 377,910 IS divisible by 3.
Check 377,910 for divisibility by \(5^2\): 377,910 IS divisible by 5 and NOT divisible by 25 (in order for a number to be divisible by 25, its last two digits must be 00, 25, 50, or 75, so 377,910 is NOT divisible by 25). Thus \(x\) must have 5 as its factor.
Check 377,910 for divisibility by 11: \((7 + 9 + 0) - (3 + 7 + 1) = 5\), so 377,910 is NOT divisible by 11 (in order for a number to be divisible by 11, the difference between the sums of the alternate digits of the given number must be either 0 or divisible by 11), thus \(x\) must have 11 as its factor.
Therefore, the least value of \(x\) is \(2*5*11=110\).
Answer: D
If the product of a positive integer \(x\) and 377,910 is divisible by 3,300, what is the least possible value of \(x\)?
A. \(10\)
B. \(11\)
C. \(55\)
D. \(110\)
E. \(330\)
Given: \(\frac{377,910*x}{3,300}=integer\).
Factorize the divisor: \(3,300=2^2*3*5^2*11\).
Check 377,910 for divisibility by \(2^2\): 377,910 IS divisible by 2 and NOT divisible by \(2^2 = 4\) (since its last two digits, 10, is not divisible by 4). Thus \(x\) must have 2 as its factor (377,910 is divisible only by 2, so in order for \(377,910*x\) to be divisible by \(2^2\), \(x\) must have 2 as its factor);
Check 377,910 for divisibility by 3: 3 + 7 + 7 + 9 + 1 + 0 = 27, thus 377,910 IS divisible by 3.
Check 377,910 for divisibility by \(5^2\): 377,910 IS divisible by 5 and NOT divisible by 25 (in order for a number to be divisible by 25, its last two digits must be 00, 25, 50, or 75, so 377,910 is NOT divisible by 25). Thus \(x\) must have 5 as its factor.
Check 377,910 for divisibility by 11: \((7 + 9 + 0) - (3 + 7 + 1) = 5\), so 377,910 is NOT divisible by 11 (in order for a number to be divisible by 11, the difference between the sums of the alternate digits of the given number must be either 0 or divisible by 11), thus \(x\) must have 11 as its factor.
Therefore, the least value of \(x\) is \(2*5*11=110\).
Answer: D
12 Jul 2016, 04:08
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I think i did it with fewer steps, but maybe i was lucky? Hopefully someone can point it out.
-> 377,910/3,300
-> 37,791/330 (cancel out zeros)
-> ^ not further divisible by 10. Therefore, x must have 10.
-> 37,791/33 (divisible by 3?Yes!)
-> 12,597/11 (divisible by 11? No)
-> x must have at least 10 and 11.
-> 110
-> 377,910/3,300
-> 37,791/330 (cancel out zeros)
-> ^ not further divisible by 10. Therefore, x must have 10.
-> 37,791/33 (divisible by 3?Yes!)
-> 12,597/11 (divisible by 11? No)
-> x must have at least 10 and 11.
-> 110
