Divisibility and Multiples | IT

Question 29

Fill in the following table. Find a quick way to do this?

Solution

Understand the Question

To quickly determine divisibility without performing long division, we use divisibility rules:

2: Last digit is even ( $0, 2, 4, 6, 8$ ).
3: Sum of digits is divisible by $3$ .
4: Number formed by the last two digits is divisible by $4$ .
5: Last digit is $0$ or $5$ .
6: Number is divisible by both $2$ and $3$ .
8: Number formed by the last three digits is divisible by $8$ .
9: Sum of digits is divisible by $9$ .
10: Last digit is $0$ .
11: Alternating sum of digits (starting from units place: $+ - + - \dots$ ) is divisible by $11$ (including $0$ ).

Step 1 · Apply Divisibility Rules to Each Number

Question diagram 1. For 128:

2: Last digit is $8$ (even) $\rightarrow$ Yes
3: Sum of digits $= 1 + 2 + 8 = 11$ (not divisible by $3$ ) $\rightarrow$ No
4: Last two digits $= 28$ ( $28 \div 4 = 7$ ) $\rightarrow$ Yes
5: Last digit is $8$ $\rightarrow$ No
6: Not divisible by $3$ $\rightarrow$ No
8: Last three digits $= 128$ ( $128 \div 8 = 16$ ) $\rightarrow$ Yes
9: Sum of digits $= 11$ (not divisible by $9$ ) $\rightarrow$ No
10: Last digit is $8$ $\rightarrow$ No
11: Alternating sum $= 8 - 2 + 1 = 7$ (not divisible by $11$ ) $\rightarrow$ No

2. For 990:

2: Last digit is $0$ (even) $\rightarrow$ Yes
3: Sum of digits $= 9 + 9 + 0 = 18$ ( $18 \div 3 = 6$ ) $\rightarrow$ Yes
4: Last two digits $= 90$ ( $90 \div 4 = 22.5$ ) $\rightarrow$ No
5: Last digit is $0$ $\rightarrow$ Yes
6: Divisible by both $2$ and $3$ $\rightarrow$ Yes
8: Last three digits $= 990$ ( $990 \div 8 = 123.75$ ) $\rightarrow$ No
9: Sum of digits $= 18$ ( $18 \div 9 = 2$ ) $\rightarrow$ Yes
10: Last digit is $0$ $\rightarrow$ Yes
11: Alternating sum $= 0 - 9 + 9 = 0$ ( $0 \div 11 = 0$ ) $\rightarrow$ Yes

3. For 1586:

2: Last digit is $6$ (even) $\rightarrow$ Yes
3: Sum of digits $= 1 + 5 + 8 + 6 = 20$ (not divisible by $3$ ) $\rightarrow$ No
4: Last two digits $= 86$ ( $86 \div 4 = 21.5$ ) $\rightarrow$ No
5: Last digit is $6$ $\rightarrow$ No
6: Not divisible by $3$ $\rightarrow$ No
8: Last three digits $= 586$ ( $586 \div 8 = 73.25$ ) $\rightarrow$ No
9: Sum of digits $= 20$ (not divisible by $9$ ) $\rightarrow$ No
10: Last digit is $6$ $\rightarrow$ No
11: Alternating sum $= 6 - 8 + 5 - 1 = 2$ (not divisible by $11$ ) $\rightarrow$ No

4. For 275:

2: Last digit is $5$ (odd) $\rightarrow$ No
3: Sum of digits $= 2 + 7 + 5 = 14$ (not divisible by $3$ ) $\rightarrow$ No
4: Last two digits $= 75$ ( $75 \div 4 = 18.75$ ) $\rightarrow$ No
5: Last digit is $5$ $\rightarrow$ Yes
6: Not divisible by $2$ $\rightarrow$ No
8: Last three digits $= 275$ ( $275 \div 8 = 34.375$ ) $\rightarrow$ No
9: Sum of digits $= 14$ (not divisible by $9$ ) $\rightarrow$ No
10: Last digit is $5$ $\rightarrow$ No
11: Alternating sum $= 5 - 7 + 2 = 0$ ( $0 \div 11 = 0$ ) $\rightarrow$ Yes

5. For 6686:

2: Last digit is $6$ (even) $\rightarrow$ Yes
3: Sum of digits $= 6 + 6 + 8 + 6 = 26$ (not divisible by $3$ ) $\rightarrow$ No
4: Last two digits $= 86$ ( $86 \div 4 = 21.5$ ) $\rightarrow$ No
5: Last digit is $6$ $\rightarrow$ No
6: Not divisible by $3$ $\rightarrow$ No
8: Last three digits $= 686$ ( $686 \div 8 = 85.75$ ) $\rightarrow$ No
9: Sum of digits $= 26$ (not divisible by $9$ ) $\rightarrow$ No
10: Last digit is $6$ $\rightarrow$ No
11: Alternating sum $= 6 - 8 + 6 - 6 = -2$ (not divisible by $11$ ) $\rightarrow$ No

6. For 639210:

2: Last digit is $0$ (even) $\rightarrow$ Yes
3: Sum of digits $= 6 + 3 + 9 + 2 + 1 + 0 = 21$ ( $21 \div 3 = 7$ ) $\rightarrow$ Yes
4: Last two digits $= 10$ ( $10 \div 4 = 2.5$ ) $\rightarrow$ No
5: Last digit is $0$ $\rightarrow$ Yes
6: Divisible by both $2$ and $3$ $\rightarrow$ Yes
8: Last three digits $= 210$ ( $210 \div 8 = 26.25$ ) $\rightarrow$ No
9: Sum of digits $= 21$ (not divisible by $9$ ) $\rightarrow$ No
10: Last digit is $0$ $\rightarrow$ Yes
11: Alternating sum $= 0 - 1 + 2 - 9 + 3 - 6 = -11$ ( $-11 \div 11 = -1$ ) $\rightarrow$ Yes

7. For 429714:

2: Last digit is $4$ (even) $\rightarrow$ Yes
3: Sum of digits $= 4 + 2 + 9 + 7 + 1 + 4 = 27$ ( $27 \div 3 = 9$ ) $\rightarrow$ Yes
4: Last two digits $= 14$ ( $14 \div 4 = 3.5$ ) $\rightarrow$ No
5: Last digit is $4$ $\rightarrow$ No
6: Divisible by both $2$ and $3$ $\rightarrow$ Yes
8: Last three digits $= 714$ ( $714 \div 8 = 89.25$ ) $\rightarrow$ No
9: Sum of digits $= 27$ ( $27 \div 9 = 3$ ) $\rightarrow$ Yes
10: Last digit is $4$ $\rightarrow$ No
11: Alternating sum $= 4 - 1 + 7 - 9 + 2 - 4 = -1$ (not divisible by $11$ ) $\rightarrow$ No

8. For 2856:

2: Last digit is $6$ (even) $\rightarrow$ Yes
3: Sum of digits $= 2 + 8 + 5 + 6 = 21$ ( $21 \div 3 = 7$ ) $\rightarrow$ Yes
4: Last two digits $= 56$ ( $56 \div 4 = 14$ ) $\rightarrow$ Yes
5: Last digit is $6$ $\rightarrow$ No
6: Divisible by both $2$ and $3$ $\rightarrow$ Yes
8: Last three digits $= 856$ ( $856 \div 8 = 107$ ) $\rightarrow$ Yes
9: Sum of digits $= 21$ (not divisible by $9$ ) $\rightarrow$ No
10: Last digit is $6$ $\rightarrow$ No
11: Alternating sum $= 6 - 5 + 8 - 2 = 7$ (not divisible by $11$ ) $\rightarrow$ No

9. For 3060:

2: Last digit is $0$ (even) $\rightarrow$ Yes
3: Sum of digits $= 3 + 0 + 6 + 0 = 9$ ( $9 \div 3 = 3$ ) $\rightarrow$ Yes
4: Last two digits $= 60$ ( $60 \div 4 = 15$ ) $\rightarrow$ Yes
5: Last digit is $0$ $\rightarrow$ Yes
6: Divisible by both $2$ and $3$ $\rightarrow$ Yes
8: Last three digits $= 060$ ( $60 \div 8 = 7.5$ ) $\rightarrow$ No
9: Sum of digits $= 9$ ( $9 \div 9 = 1$ ) $\rightarrow$ Yes
10: Last digit is $0$ $\rightarrow$ Yes
11: Alternating sum $= 0 - 6 + 0 - 3 = -9$ (not divisible by $11$ ) $\rightarrow$ No

10. For 406839:

2: Last digit is $9$ (odd) $\rightarrow$ No
3: Sum of digits $= 4 + 0 + 6 + 8 + 3 + 9 = 30$ ( $30 \div 3 = 10$ ) $\rightarrow$ Yes
4: Last two digits $= 39$ ( $39 \div 4 = 9.75$ ) $\rightarrow$ No
5: Last digit is $9$ $\rightarrow$ No
6: Not divisible by $2$ $\rightarrow$ No
8: Last three digits $= 839$ ( $839 \div 8 = 104.875$ ) $\rightarrow$ No
9: Sum of digits $= 30$ (not divisible by $9$ ) $\rightarrow$ No
10: Last digit is $9$ $\rightarrow$ No
11: Alternating sum $= 9 - 3 + 8 - 6 + 0 - 4 = 4$ (not divisible by $11$ ) $\rightarrow$ No

Answer

$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline\text{Number} & \text{Divisible by 2} & \text{Divisible by 3} & \text{Divisible by 4} & \text{Divisible by 5} & \text{Divisible by 6} & \text{Divisible by 8} & \text{Divisible by 9} & \text{Divisible by 10} & \text{Divisible by 11} \\\hline 128 & \text{Yes} & \text{No} & \text{Yes} & \text{No} & \text{No} & \text{Yes} & \text{No} & \text{No} & \text{No} \\\hline 990 & \text{Yes} & \text{Yes} & \text{No} & \text{Yes} & \text{Yes} & \text{No} & \text{Yes} & \text{Yes} & \text{Yes} \\\hline 1586 & \text{Yes} & \text{No} & \text{No} & \text{No} & \text{No} & \text{No} & \text{No} & \text{No} & \text{No} \\\hline 275 & \text{No} & \text{No} & \text{No} & \text{Yes} & \text{No} & \text{No} & \text{No} & \text{No} & \text{Yes} \\\hline 6686 & \text{Yes} & \text{No} & \text{No} & \text{No} & \text{No} & \text{No} & \text{No} & \text{No} & \text{No} \\\hline 639210 & \text{Yes} & \text{Yes} & \text{No} & \text{Yes} & \text{Yes} & \text{No} & \text{No} & \text{Yes} & \text{Yes} \\\hline 429714 & \text{Yes} & \text{Yes} & \text{No} & \text{No} & \text{Yes} & \text{No} & \text{Yes} & \text{No} & \text{No} \\\hline 2856 & \text{Yes} & \text{Yes} & \text{Yes} & \text{No} & \text{Yes} & \text{Yes} & \text{No} & \text{No} & \text{No} \\\hline 3060 & \text{Yes} & \text{Yes} & \text{Yes} & \text{Yes} & \text{Yes} & \text{No} & \text{Yes} & \text{Yes} & \text{No} \\\hline 406839 & \text{No} & \text{Yes} & \text{No} & \text{No} & \text{No} & \text{No} & \text{No} & \text{No} & \text{No} \\\hline\end{array}$

Common Mistakes

Divisibility by 6: A number must be divisible by both $2$ and $3$ . If it fails divisibility by either, it is not divisible by $6$ .
Divisibility by 9 vs 3: A number divisible by $3$ is not automatically divisible by 9 (e.g., sum $= 21$ or $30$ is divisible by $3$ , but not by $9$ ).
Divisibility by 11 with Sum of 0: An alternating sum equal to $0$ is divisible by $11$ ( $0 \div 11 = 0$ ), so the number is divisible by $11$ .

More on Divisibility Shortcuts

Divisibility Shortcuts for Other Numbers

How can we find out if a number is divisible by 6?

Q31

Will checking its divisibility by its factors 2 and 3 work? Use the shortcuts for 2 and 3 on these numbers and divide each number by 6 to verify— 38, 225, 186, 64.

Q32

How about checking divisibility by 24? Will checking the divisibility by its factors, 4 and 6, work? Why or why not?

Q33

What property do you think this digital root will have? Recall that we did this while finding the divisibility shortcut for 9.

Q34

Between the numbers 600 and 700, which numbers have the digital root: (i) 5, (ii) 7, (iii) 3?

Q35

Write the digital roots of any 12 consecutive numbers. What do you observe?

Q36

Now, find the digital roots of some consecutive multiples of (i) 3, (ii) 4, and (iii) 6.

Q37

What are the digital roots of numbers that are 1 more than a multiple of 6? What do you notice?

Try to explain the patterns noticed.

Q38

I’m made of digits, each tiniest and odd, No shared ground with root #1—how odd!

My digits count, their sum, my root— All point to one bold number’s pursuit— The largest odd single-digit I proudly claim.

What’s my number? What’s my name?

Q39

Solve the cryptarithms given below:

Q40

Try this now: $\text{GH} \times \text{H} = 9\text{K}$ .

This means a 2-digit number multiplied by a 1-digit number gives another 2-digit number in the 90s. Observe the letters corresponding to the units digits in this cryptarithm. Pick the solution to this question from the options given below: $11 \times 9 = 99$ , $12 \times 8 = 96$ , $46 \times 2 = 92$ , $24 \times 4 = 96$ , $47 \times 2 = 94$ , $31 \times 3 = 93$ , $16 \times 6 = 96$ .

Q41

Solve the following:

(i) $\text{UT} \times 3 = \text{PUT}$

(ii) $\text{AB} \times 5 = \text{BC}$

(iii) $\text{L2N} \times 2 = \text{2NP}$

(iv) $\text{XY} \times 4 = \text{ZX}$

(v) $\text{PP} \times \text{QQ} = \text{PRP}$

(vi) $\text{JK} \times 6 = \text{KKK}$

Question 29

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