Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2017 Grade 8 Number Sense
Problem
Hint
1) Insert parentheses to make the following equation true.
    60 + 12 ÷ 2 - 3 x 22 = 0
  1. To sum to 0, the expressions to either side of that negative sign must be equal.
  2. Try various combinations of parentheses that make these 2 expressions equal =


    _______________________________
2) Rewrite 0.132 as a fraction in lowest terms.
 The explanation for repeating decimals shows you how to do this.
  1. Call the repeating decimal X
  2. Multiply the number by a power of 10 to get the repeating part to the left of the decimal:
    ____________________________
  3. If there are leading non-repeating decimals, get them also to the left of the decimal point by multiplying the number by a power of 10 to get the repeating part to the right of the decimal:


        _____________________
  4. Subtract this second value from the larger value:


    = ___________
  5. ____X - ___X = ____ (the result of the above subtraction)


    X = __________

Problem
Hint
3) What is the one's place of 4355?

 This kind of problem relies on the fact that some numbers have repeating one's place digits when taken to increasing powers. Keep taking powers of 43 (2,3,4, ...) until you notice a pattern. The one's digit repeats every ____ power. Divide the power you want by the number of repeating digits. The remainder will be the number of the digit in the repeating sequence.








The one's digit for 4355 = _____
4) What numbers between 1 and 100 have exactly 5 factors?
  1. If a number has exactly 5 factors, then there are only 3 factors that are not 1 and the number itself.
  2. Square numbers have an ODD number of factors. This is because factors are always in pairs, but in a square, one of the factors is multiplied by itself and it is only counted once. (the square root). Therefore the numbers that have 5 factors will be squares.
  3. The numbers that have 5 factors will be "squares of squares" or 4th powers of some basal number.
  4. The only numbers which have 3 factors, other than 1 and the number itself, that are less than 100 are

    _________________
5) Given the following sequence, find what the 10th term in the sequence is. Express your answer in a simplified form.
Solution: Starting at the first term, perform the addition, leaving the result in the form of an improper fraction. Keep substituting this value in the subsequent terms and simplifying. You should notice a pattern.

  1. 1 + 2⁄3 = ______
  2. 1 + 1/______ = ______

  3. 1 + 1/_______ = _______

  4. 1 + 1/_________= ________
Calling the fraction A / B, each successive element is _________ of the previous element, so, extending this to the 10th term:
Term=5678910