Question 4

We will look for unique mathematical properties for each number in the orange boxes.

Step 1 — First Grid Numbers

Let us look at the numbers in the orange boxes of the first grid. These numbers are 105, 20, 30, 28, 125, and 18.

105: It is an odd number. It is a multiple of 3, 5, and 7. We check its factors. $105 \div 3 = 35$ $105 \div 5 = 21$ $105 \div 7 = 15$

$\boxed{\text{105 is a multiple of 3, 5, and 7}}$ No other number in this list is a multiple of all three.

20: It is an even number. It is a multiple of 4 and 5. We check its factors. $20 \div 4 = 5$ $20 \div 5 = 4$

$\boxed{\text{20 is a multiple of 4 and 5}}$ No other number in this list is a multiple of both 4 and 5.

30: It is an even number. It is a multiple of 2, 3, and 5. We check its factors. $30 \div 2 = 15$ $30 \div 3 = 10$ $30 \div 5 = 6$

$\boxed{\text{30 is a multiple of 2, 3, and 5}}$ No other number in this list is a multiple of all three.

28: It is an even number. It is a multiple of 4 and 7. We check its factors. $28 \div 4 = 7$ $28 \div 7 = 4$

$\boxed{\text{28 is a multiple of 4 and 7}}$ No other number in this list is a multiple of both 4 and 7.

125: It is an odd number. It is a perfect cube. We find its factors. $5 \times 5 \times 5$ $= 25 \times 5$ $= 125$

$\boxed{\text{125 is a perfect cube}}$ No other number in this list is a perfect cube.

18: It is an even number. It is a multiple of 9. We check its factors. $18 \div 9 = 2$

$\boxed{\text{18 is a multiple of 9}}$ No other number in this list is a multiple of 9.

Step 2 — Second Grid Numbers

Let us look at the numbers in the orange boxes of the second grid. These numbers are 8, 105, 70, 30, 70, and 28.

8: It is an even number. It is a perfect cube. We find its factors. $2 \times 2 \times 2$ $= 4 \times 2$ $= 8$

$\boxed{\text{8 is a perfect cube}}$ No other number in this list is a perfect cube.

105: It is an odd number. It is a multiple of 3, 5, and 7. We check its factors. $105 \div 3 = 35$ $105 \div 5 = 21$ $105 \div 7 = 15$

$\boxed{\text{105 is a multiple of 3, 5, and 7}}$ No other number in this list is a multiple of all three.

70: It is an even number. It is a multiple of 10 and 7. We check its factors. $70 \div 10 = 7$ $70 \div 7 = 10$

$\boxed{\text{70 is a multiple of 10 and 7}}$ No other number in this list is a multiple of both 10 and 7. This number also appears twice in the given orange cells.

30: It is an even number. It is a multiple of 2, 3, and 5. We check its factors. $30 \div 2 = 15$ $30 \div 3 = 10$ $30 \div 5 = 6$

$\boxed{\text{30 is a multiple of 2, 3, and 5}}$ No other number in this list is a multiple of all three.

28: It is an even number. It is a multiple of 4 and 7. We check its factors. $28 \div 4 = 7$ $28 \div 7 = 4$

$\boxed{\text{28 is a multiple of 4 and 7}}$ No other number in this list is a multiple of both 4 and 7.

Answer

(i) For the first grid, 105 is special because it is an odd number and a multiple of 3, 5, and 7. (ii) For the first grid, 20 is special because it is an even number and a multiple of 4 and 5. (iii) For the first grid, 30 is special because it is an even number and a multiple of 2, 3, and 5. (iv) For the first grid, 28 is special because it is an even number and a multiple of 4 and 7. (v) For the first grid, 125 is special because it is an odd number and a perfect cube. (vi) For the first grid, 18 is special because it is an even number and a multiple of 9. (vii) For the second grid, 8 is special because it is an even number and a perfect cube. (viii) For the second grid, 105 is special because it is an odd number and a multiple of 3, 5, and 7. (ix) For the second grid, 70 is special because it is an even number, a multiple of 10 and 7, and it appears twice. (x) For the second grid, 30 is special because it is an even number and a multiple of 2, 3, and 5. (xi) For the second grid, 28 is special because it is an even number and a multiple of 4 and 7.