Divisibility test worksheet for practicing divisibility rules.

To solve the problem, we need to use the divisibility rules for each number at the top of the columns (2, 3, 4, 5, 6, 9, and 10) and check if the numbers in the rows (90, 88, 75, 16, 87, 27, 70, 84, 99, and 52) are divisible by these numbers. Let's go through each row and column step by step.

Divisibility Rules Recap:

1. Divisible by 2: The number is even (last digit is 0, 2, 4, 6, or 8).
2. Divisible by 3: The sum of the digits is divisible by 3.
3. Divisible by 4: The last two digits form a number divisible by 4.
4. Divisible by 5: The last digit is 0 or 5.
5. Divisible by 6: The number is divisible by both 2 and 3.
6. Divisible by 9: The sum of the digits is divisible by 9.
7. Divisible by 10: The last digit is 0.

Step-by-Step Solution:

#### Row 1: 90
- 2: 90 is even. YES
- 3: Sum of digits = 9 + 0 = 9 (divisible by 3). YES
- 4: Last two digits = 90 (not divisible by 4). NO
- 5: Last digit = 0. YES
- 6: Divisible by both 2 and 3. YES
- 9: Sum of digits = 9 + 0 = 9 (divisible by 9). YES
- 10: Last digit = 0. YES

#### Row 2: 88
- 2: 88 is even. YES
- 3: Sum of digits = 8 + 8 = 16 (not divisible by 3). NO
- 4: Last two digits = 88 (divisible by 4). YES
- 5: Last digit ≠ 0 or 5. NO
- 6: Not divisible by 3. NO
- 9: Sum of digits = 8 + 8 = 16 (not divisible by 9). NO
- 10: Last digit ≠ 0. NO

#### Row 3: 75
- 2: 75 is odd. NO
- 3: Sum of digits = 7 + 5 = 12 (divisible by 3). YES
- 4: Last two digits = 75 (not divisible by 4). NO
- 5: Last digit = 5. YES
- 6: Not divisible by 2. NO
- 9: Sum of digits = 7 + 5 = 12 (not divisible by 9). NO
- 10: Last digit ≠ 0. NO

#### Row 4: 16
- 2: 16 is even. YES
- 3: Sum of digits = 1 + 6 = 7 (not divisible by 3). NO
- 4: Last two digits = 16 (divisible by 4). YES
- 5: Last digit ≠ 0 or 5. NO
- 6: Not divisible by 3. NO
- 9: Sum of digits = 1 + 6 = 7 (not divisible by 9). NO
- 10: Last digit ≠ 0. NO

#### Row 5: 87
- 2: 87 is odd. NO
- 3: Sum of digits = 8 + 7 = 15 (divisible by 3). YES
- 4: Last two digits = 87 (not divisible by 4). NO
- 5: Last digit ≠ 0 or 5. NO
- 6: Not divisible by 2. NO
- 9: Sum of digits = 8 + 7 = 15 (not divisible by 9). NO
- 10: Last digit ≠ 0. NO

#### Row 6: 27
- 2: 27 is odd. NO
- 3: Sum of digits = 2 + 7 = 9 (divisible by 3). YES
- 4: Last two digits = 27 (not divisible by 4). NO
- 5: Last digit ≠ 0 or 5. NO
- 6: Not divisible by 2. NO
- 9: Sum of digits = 2 + 7 = 9 (divisible by 9). YES
- 10: Last digit ≠ 0. NO

#### Row 7: 70
- 2: 70 is even. YES
- 3: Sum of digits = 7 + 0 = 7 (not divisible by 3). NO
- 4: Last two digits = 70 (not divisible by 4). NO
- 5: Last digit = 0. YES
- 6: Not divisible by 3. NO
- 9: Sum of digits = 7 + 0 = 7 (not divisible by 9). NO
- 10: Last digit = 0. YES

#### Row 8: 84
- 2: 84 is even. YES
- 3: Sum of digits = 8 + 4 = 12 (divisible by 3). YES
- 4: Last two digits = 84 (divisible by 4). YES
- 5: Last digit ≠ 0 or 5. NO
- 6: Divisible by both 2 and 3. YES
- 9: Sum of digits = 8 + 4 = 12 (not divisible by 9). NO
- 10: Last digit ≠ 0. NO

#### Row 9: 99
- 2: 99 is odd. NO
- 3: Sum of digits = 9 + 9 = 18 (divisible by 3). YES
- 4: Last two digits = 99 (not divisible by 4). NO
- 5: Last digit ≠ 0 or 5. NO
- 6: Not divisible by 2. NO
- 9: Sum of digits = 9 + 9 = 18 (divisible by 9). YES
- 10: Last digit ≠ 0. NO

#### Row 10: 52
- 2: 52 is even. YES
- 3: Sum of digits = 5 + 2 = 7 (not divisible by 3). NO
- 4: Last two digits = 52 (divisible by 4). YES
- 5: Last digit ≠ 0 or 5. NO
- 6: Not divisible by 3. NO
- 9: Sum of digits = 5 + 2 = 7 (not divisible by 9). NO
- 10: Last digit ≠ 0. NO

Final Answer:

\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
& 2 & 3 & 4 & 5 & 6 & 9 & 10 \\
\hline
90 & \text{YES} & \text{YES} & \text{NO} & \text{YES} & \text{YES} & \text{YES} & \text{YES} \\
\hline
88 & \text{YES} & \text{NO} & \text{YES} & \text{NO} & \text{NO} & \text{NO} & \text{NO} \\
\hline
75 & \text{NO} & \text{YES} & \text{NO} & \text{YES} & \text{NO} & \text{NO} & \text{NO} \\
\hline
16 & \text{YES} & \text{NO} & \text{YES} & \text{NO} & \text{NO} & \text{NO} & \text{NO} \\
\hline
87 & \text{NO} & \text{YES} & \text{NO} & \text{NO} & \text{NO} & \text{NO} & \text{NO} \\
\hline
27 & \text{NO} & \text{YES} & \text{NO} & \text{NO} & \text{NO} & \text{YES} & \text{NO} \\
\hline
70 & \text{YES} & \text{NO} & \text{NO} & \text{YES} & \text{NO} & \text{NO} & \text{YES} \\
\hline
84 & \text{YES} & \text{YES} & \text{YES} & \text{NO} & \text{YES} & \text{NO} & \text{NO} \\
\hline
99 & \text{NO} & \text{YES} & \text{NO} & \text{NO} & \text{NO} & \text{YES} & \text{NO} \\
\hline
52 & \text{YES} & \text{NO} & \text{YES} & \text{NO} & \text{NO} & \text{NO} & \text{NO} \\
\hline
\end{array}
\]

\boxed{
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
& 2 & 3 & 4 & 5 & 6 & 9 & 10 \\
\hline
90 & \text{YES} & \text{YES} & \text{NO} & \text{YES} & \text{YES} & \text{YES} & \text{YES} \\
\hline
88 & \text{YES} & \text{NO} & \text{YES} & \text{NO} & \text{NO} & \text{NO} & \text{NO} \\
\hline
75 & \text{NO} & \text{YES} & \text{NO} & \text{YES} & \text{NO} & \text{NO} & \text{NO} \\
\hline
16 & \text{YES} & \text{NO} & \text{YES} & \text{NO} & \text{NO} & \text{NO} & \text{NO} \\
\hline
87 & \text{NO} & \text{YES} & \text{NO} & \text{NO} & \text{NO} & \text{NO} & \text{NO} \\
\hline
27 & \text{NO} & \text{YES} & \text{NO} & \text{NO} & \text{NO} & \text{YES} & \text{NO} \\
\hline
70 & \text{YES} & \text{NO} & \text{NO} & \text{YES} & \text{NO} & \text{NO} & \text{YES} \\
\hline
84 & \text{YES} & \text{YES} & \text{YES} & \text{NO} & \text{YES} & \text{NO} & \text{NO} \\
\hline
99 & \text{NO} & \text{YES} & \text{NO} & \text{NO} & \text{NO} & \text{YES} & \text{NO} \\
\hline
52 & \text{YES} & \text{NO} & \text{YES} & \text{NO} & \text{NO} & \text{NO} & \text{NO} \\
\hline
\end{array}
}