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1
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A positive integer n is said to be "prime-saturated" if the product of all the different positive prime factors of n is less than the square root of n. What is the greatest two-digit prime-saturated integer?
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2
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A photographer will arrange 6 people of 6 different heights for photograph by placing them in two rows of three so that each person in the first row is standing in front of someone in the second row. The heights of the people within each row must increase from left to right, and each person in the second row must be taller than the person standing in front of him or her. How many such arrangements of the 6 people are possible?
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3
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A box contains 100 balls, numbered from 1 to 100. If three balls are selected at random and with replacement from the box, what is the probability that the sum of the three numbers on the balls selected from the box will be odd?
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4
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 A rectangular yard (white area) is surrounded by a hedge (shaded region) with a length of 15 meters and a width of 12 meters, as shown in the figure above. The area of the yard is equal to the area of the hedge. If the ratio of the length of the hedge to the width of the hedge is the same as the ratio of the length of the yard to the width of the yard, what is the width, in meters, of the yard?
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5
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$$x^{8}-y^{8}=$$
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6
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If $$\sqrt{{3}-{2}{x}}=\sqrt{2x}+{1}$$,then $${4}{x}^{2}=$$
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7
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When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7, the remainder is 4. When positive integer y is divided by 5, the remainder is 3; and when y is divided by 7, the remainder is 4. If x>y, which of the following must be a factor of x - y?
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10
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A positive integer with exactly two different divisors greater than 1 must be
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11
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If x and y are different prime numbers, each greater than 2, which of the following must betrue?I $${x}+{y} \neq{91} $$II $$x - y$$ is an even integer.III $$\frac{x}{y}$$is not an integer.
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12
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How many positive odd factors larger than 1 does 210 have?
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13
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If a positive integer n is divisible by both 5 and 7, then n must also be divisible by which of the following?I 12II 35III 70
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14
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A positive integer has a remainder of 5 when divided by 10 and a remainder of 3 when divided by 8. What is the general formula for this number?
(Note: k is a natural number.)
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15
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If x and y are positive integers such that the remainder when x is divided by 5 is 2 and the remainder when y is divided by 5 is 3, what is the remainder when xy is divided by 5?
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16
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If n is a multiple of 3, what is the remainder when 15n is divided by 6?
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17
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What is the remainder when $$3^{50}$$ is divided by 4？
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18
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If n is a positive integer, what is the remainder when $${3}^{8n+3}+2$$ is divided by 5?
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19
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If n is a positive integer, what could the unit digit of n(n+1)(n+2)(n+3)(n+4)?
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20
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Which of the following fractions has a decimal equivalent that is a terminating decimal?
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21
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How many of the integers that satisfy the inequality (y+1)(y+2)(y+4)≥0 are less than 4?
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