Divisibility Rules Worksheets - Math Worksheets - MathsDiary.com - Free Printable

Problem Explanation:

The task is to determine which numbers (on the left column) are divisible by the given divisors (2, 3, 4, 5, 6, 9, and 10). A checkmark (√) should be placed under each divisor that divides the number without leaving a remainder.

Solution Approach:

To solve this, we will check each number for divisibility by each of the given divisors using the following rules:
1. Divisibility by 2: The number must be even (last digit is 0, 2, 4, 6, or 8).
2. Divisibility by 3: The sum of the digits of the number must be divisible by 3.
3. Divisibility by 4: The last two digits of the number must form a number divisible by 4.
4. Divisibility by 5: The last digit of the number must be 0 or 5.
5. Divisibility by 6: The number must be divisible by both 2 and 3.
6. Divisibility by 9: The sum of the digits of the number must be divisible by 9.
7. Divisibility by 10: The last digit of the number must be 0.

Step-by-Step Analysis:

#### 1. 45
- 2: Not even → Not divisible.
- 3: Sum of digits = 4 + 5 = 9 (divisible by 3) → Divisible.
- 4: Last two digits = 45 (not divisible by 4) → Not divisible.
- 5: Last digit = 5 → Divisible.
- 6: Not divisible by 2 → Not divisible.
- 9: Sum of digits = 9 (divisible by 9) → Divisible.
- 10: Last digit ≠ 0 → Not divisible.

Result: √ under 3, 5, and 9.

#### 2. 369
- 2: Not even → Not divisible.
- 3: Sum of digits = 3 + 6 + 9 = 18 (divisible by 3) → Divisible.
- 4: Last two digits = 69 (not divisible by 4) → Not divisible.
- 5: Last digit ≠ 5 or 0 → Not divisible.
- 6: Not divisible by 2 → Not divisible.
- 9: Sum of digits = 18 (divisible by 9) → Divisible.
- 10: Last digit ≠ 0 → Not divisible.

Result: √ under 3 and 9.

#### 3. 7,870
- 2: Even → Divisible.
- 3: Sum of digits = 7 + 8 + 7 + 0 = 22 (not divisible by 3) → Not divisible.
- 4: Last two digits = 70 (not divisible by 4) → Not divisible.
- 5: Last digit = 0 → Divisible.
- 6: Not divisible by 3 → Not divisible.
- 9: Sum of digits = 22 (not divisible by 9) → Not divisible.
- 10: Last digit = 0 → Divisible.

Result: √ under 2, 5, and 10.

#### 4. 1,976
- 2: Even → Divisible.
- 3: Sum of digits = 1 + 9 + 7 + 6 = 23 (not divisible by 3) → Not divisible.
- 4: Last two digits = 76 (divisible by 4) → Divisible.
- 5: Last digit ≠ 5 or 0 → Not divisible.
- 6: Not divisible by 3 → Not divisible.
- 9: Sum of digits = 23 (not divisible by 9) → Not divisible.
- 10: Last digit ≠ 0 → Not divisible.

Result: √ under 2 and 4.

#### 5. 6,003
- 2: Not even → Not divisible.
- 3: Sum of digits = 6 + 0 + 0 + 3 = 9 (divisible by 3) → Divisible.
- 4: Last two digits = 03 (not divisible by 4) → Not divisible.
- 5: Last digit ≠ 5 or 0 → Not divisible.
- 6: Not divisible by 2 → Not divisible.
- 9: Sum of digits = 9 (divisible by 9) → Divisible.
- 10: Last digit ≠ 0 → Not divisible.

Result: √ under 3 and 9.

#### 6. 136
- 2: Even → Divisible.
- 3: Sum of digits = 1 + 3 + 6 = 10 (not divisible by 3) → Not divisible.
- 4: Last two digits = 36 (divisible by 4) → Divisible.
- 5: Last digit ≠ 5 or 0 → Not divisible.
- 6: Not divisible by 3 → Not divisible.
- 9: Sum of digits = 10 (not divisible by 9) → Not divisible.
- 10: Last digit ≠ 0 → Not divisible.

Result: √ under 2 and 4.

#### 7. 1,674
- 2: Even → Divisible.
- 3: Sum of digits = 1 + 6 + 7 + 4 = 18 (divisible by 3) → Divisible.
- 4: Last two digits = 74 (not divisible by 4) → Not divisible.
- 5: Last digit ≠ 5 or 0 → Not divisible.
- 6: Divisible by both 2 and 3 → Divisible.
- 9: Sum of digits = 18 (divisible by 9) → Divisible.
- 10: Last digit ≠ 0 → Not divisible.

Result: √ under 2, 3, 6, and 9.

#### 8. 35,496
- 2: Even → Divisible.
- 3: Sum of digits = 3 + 5 + 4 + 9 + 6 = 27 (divisible by 3) → Divisible.
- 4: Last two digits = 96 (divisible by 4) → Divisible.
- 5: Last digit ≠ 5 or 0 → Not divisible.
- 6: Divisible by both 2 and 3 → Divisible.
- 9: Sum of digits = 27 (divisible by 9) → Divisible.
- 10: Last digit ≠ 0 → Not divisible.

Result: √ under 2, 3, 4, 6, and 9.

Final Answer:

\[
\boxed{
\begin{array}{c|ccccccc}
& 2 & 3 & 4 & 5 & 6 & 9 & 10 \\
\hline
45 & & \checkmark & & \checkmark & & \checkmark & \\
369 & & \checkmark & & & & \checkmark & \\
7,870 & \checkmark & & & \checkmark & & & \checkmark \\
1,976 & \checkmark & & \checkmark & & & & \\
6,003 & & \checkmark & & & & \checkmark & \\
136 & \checkmark & & \checkmark & & & & \\
1,674 & \checkmark & \checkmark & & & \checkmark & \checkmark & \\
35,496 & \checkmark & \checkmark & \checkmark & & \checkmark & \checkmark & \\
\end{array}
}
\]