1) If a class of children is separated into groups of 5 children, 2 children will be left over. If the class is separated into groups of 6 children, 3 children will be left over. What is the smallest number of children the class could have?
2) There is an even number between 200 and 300 that is divisible by 5 and also by 9. What is the number? 3) A book has 500 pages numbered 1, 2, 3, and so on. How many times does the digit 1 appear in the page numbers? Solutions 1) Method 1 –
must be an odd number. Then the only possibilities for the class size are: 7, 17, 27, 37, . . Divide each by 6. The first number that has a remainder of 3 is the required number. That number is 27. Method 3  Algebra: Let A and B be numbers such that 5A + 2 = 6B + 3. Subtract 2 from each member of the equation. Then 5A = 6B + 1. Since 6B + 1 is an odd number, 5A is also odd. 5A may be 5, 15, 25, 35, 45, . . . Divide each of these numbers by 6. The first number that has a remainder of 1 is 25: 25/6 = 4 R1. So B = 4. Since 6B + 3 is the class size, and B = 4, the class size is 6 x 4 + 3 = 2. 2) Since the number is divisible by 2, 5, and 9, it is also divisible by the product 2 x 5 x 9 = 90. The multiple of 90 between 200 and 300 is 270. 3) Consider the frequency of appearance of the digit "1" in each of the places.
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