Prime Time | A

Question 4

Below are some boxes with four numbers in each box. Within each box try to say how each number is special compared to the rest. Share with your classmates and find out who else gave the same reasons as you did. Did anyone give different reasons that may not have occurred to you?

Solution

Understand the Question

In this activity, we analyze groups of numbers to find a unique mathematical property for each number that distinguishes it from the others in the set.

Unique properties can include:

Even or Odd
Multiples and Divisibility (e.g., divisible by $3$ , $5$ , $7$ , etc.)
Powers (such as being a perfect square or perfect cube)
Repeated occurrences in the grid

Step 1 · Analyze Numbers in the First Grid

Diagram 1

Analyzing the numbers 105, 20, 30, 28, 125, and 18:

105: Odd number and a multiple of $3$ , $5$ , and $7$ .

\begin{aligned} 105 \div 3 &= 35 \\ 105 \div 5 &= 21 \\ 105 \div 7 &= 15 \end{aligned}

20: Even number and a multiple of $4$ and $5$ .

\begin{aligned} 20 \div 4 &= 5 \\ 20 \div 5 &= 4 \end{aligned}

30: Even number and a multiple of $2$ , $3$ , and $5$ .

\begin{aligned} 30 \div 2 &= 15 \\ 30 \div 3 &= 10 \\ 30 \div 5 &= 6 \end{aligned}

28: Even number and a multiple of $4$ and $7$ .

\begin{aligned} 28 \div 4 &= 7 \\ 28 \div 7 &= 4 \end{aligned}

125: Odd number and a perfect cube ( $5^3$ ).

\begin{aligned} 5 \times 5 \times 5 &= 25 \times 5 \\ &= 125 \end{aligned}

18: Even number and a multiple of $9$ . $18 \div 9 = 2$

Step 2 · Analyze Numbers in the Second Grid

Analyzing the numbers 8, 105, 70, 30, 70, and 28:

8: Even number and a perfect cube ( $2^3$ ).

\begin{aligned} 2 \times 2 \times 2 &= 4 \times 2 \\ &= 8 \end{aligned}

105: Odd number and a multiple of $3$ , $5$ , and $7$ .

\begin{aligned} 105 \div 3 &= 35 \\ 105 \div 5 &= 21 \\ 105 \div 7 &= 15 \end{aligned}

70: Even number, a multiple of $10$ and $7$ , and appears twice.

\begin{aligned} 70 \div 10 &= 7 \\ 70 \div 7 &= 10 \end{aligned}

30: Even number and a multiple of $2$ , $3$ , and $5$ .

\begin{aligned} 30 \div 2 &= 15 \\ 30 \div 3 &= 10 \\ 30 \div 5 &= 6 \end{aligned}

28: Even number and a multiple of $4$ and $7$ .

\begin{aligned} 28 \div 4 &= 7 \\ 28 \div 7 &= 4 \end{aligned}

Answer

Each number has a distinct mathematical property:

First Grid: $105$ (multiple of $3, 5, 7$ ), $20$ (multiple of $4, 5$ ), $30$ (multiple of $2, 3, 5$ ), $28$ (multiple of $4, 7$ ), $125$ (perfect cube $5^3$ ), $18$ (multiple of $9$ ).
Second Grid: $8$ (perfect cube $2^3$ ), $105$ (multiple of $3, 5, 7$ ), $70$ (multiple of $7, 10$ ; appears twice), $30$ (multiple of $2, 3, 5$ ), $28$ (multiple of $4, 7$ ).

Common Mistakes

Assuming a single unique answer: Open-ended classification problems can have multiple valid mathematical reasons (e.g., $18$ is a multiple of $9$ , but it is also the only number with a digit sum of $9$ ).
Overlooking powers: Missing special properties like being a perfect square or perfect cube ( $8 = 2^3$ , $125 = 5^3$ ).
Incomplete divisibility check: Checking only for even/odd parity without checking prime factors and composite factors.

Question 4

More questions in A

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A prime puzzle

Rules