NCERT class VI mathematics for blind and low vision students.

Is the number 38 divisible by 2? by 4? by 5?

By actually dividing 38 by these numbers we find that it is divisible by 2 but
not by 4 and by 5.

Let us see whether we can find a pattern that can tell us whether a number is divisible by 2, 3, 4, 5, 6, 8, 9, 10 or 11. Do you think such patterns can be easily seen?

Divisibility by 10 :

Charu was looking at the multiples of 10. The multiples are 10, 20, 30, 40, 50, 60, ... . She found
something common in these numbers. Can you tell what?

Each of these numbers has 0 in the ones place.

She thought of some more numbers with 0 at ones place like 100, 1000, 3200, 7010. She also found that all such numbers are divisible by 10.

She finds that if a number has 0 in the ones place then it is divisible by 10.
Can you find out the divisibility rule for 100?

Divisibility by 5 :

Mani found some interesting pattern in the numbers 5, 10, 15, 20, 25, 30, 35, ... Can you tell the pattern? Look at the units place. All these numbers have either 0 or 5 in their ones place. We know that these numbers are divisible by 5.

Mani took up some more numbers that are divisible by 5, like 105, 215, 6205, 3500. Again these numbers have either 0 or 5 in their ones places.

He tried to divide the numbers 23, 56, 97 by 5. Will he be able to do that?
Check it. He observes that a number which has either 0 or 5 in its ones place is divisible by 5, other numbers leave a remainder.

Is 1,750,125 divisible by 5?

Divisibility by 2 :

Charu observes a few multiples of 2 to be 10, 12, 14, 16... and also numbers like 2410, 4356, 1358, 2972, 5974. She finds some pattern in the ones place of these numbers. Can you tell that? These numbers have only the digits 0, 2, 4, 6, 8 in the ones place.

She divides these numbers by 2 and gets remainder 0.

She also finds that the numbers 2467, 4829 are not divisible by 2. These numbers do not have 0, 2, 4, 6 or 8 in their ones place.

Looking at these observations she concludes that a number is divisible by 2 if it has any of the digits 0, 2, 4, 6 or 8 in its ones place.

Divisibility by 3 :

Are the numbers 21, 27, 36, 54, 219 divisible by 3? Yes, they are.

Are the numbers 25, 37, 260 divisible by 3? No.

Can you see any pattern in the ones place? We cannot, because numbers with the same digit in the ones places can be divisible by 3, like 27, or may not be divisible by 3 like 17, 37. Let us now try to add the digits of 21, 36, 54 and 219. Do you observe anything special ? 2+1=3, 3+6=9, 5+4=9, 2+1+9=12. All these additions are divisible by 3.

Add the digits in 25, 37, 260. We get 2+5=7, 3+7=10, 2+6+0 = 8.
These are not divisible by 3.

We say that if the sum of the digits of a number is a multiple of 3, then the number is divisible by 3.

Is 7221 divisible by 3?

Divisibility by 6 :

Can you identify a number which is divisible by both 2 and 3? One such number is 18. Will 18 be divisible by 2×3=6? Yes, it is.

Find some more numbers like 18 and check if they are divisible by 6 also.

Can you quickly think of a number which is divisible by 2 but not by 3?

Now for a number divisible by 3 but not by 2, one example is 27. Is 27 divisible by 6? No. Try to find numbers like 27.

From these observations we conclude that if a number is divisible by 2 and 3 both then it is divisible by 6 also.

Divisibility by 4 :

Can you quickly give five 3-digit numbers divisible by 4? One such number is 212. Think of such 4-digit numbers. One example is 1936.

Observe the number formed by the ones and tens places of 212. It is 12;
which is divisible by 4. For 1936 it is 36, again divisible by 4.
Try the exercise with other such numbers,
for example with
4612;
3516;
9532.
Is the number 286 divisible by 4? No. Is 86 divisible by 4? No.

So, we see that a number with 3 or more digits is divisible by 4 if the number formed by its last two digits (i.e. ones and tens) is divisible by 4.

Check this rule by taking ten more examples.

Divisibility for 1 or 2 digit numbers by 4 has to be checked by actual division.

Divisibility by 8 :

Are the numbers 1000, 2104, 1416 divisible by 8?

You can check that they are divisible by 8. Let us try to see the pattern.

Look at the digits at ones, tens and hundreds place of these numbers. These are 000, 104 and 416 respectively. These too are divisible by 8. Find some more numbers in which the number formed by the digits at units, tens and hundreds place (i.e. last 3 digits) is divisible by 8. For example, 9216, 8216, 7216, 10216, 9995216 etc. You will find that the numbers themselves are divisible by 8.

We find that a number with 4 or more digits is divisible by 8, if the number formed by the last three digits is divisible by 8.

Is 73512 divisible by 8?
The divisibility for numbers with 1, 2 or 3 digits by 8 has to be checked by actual division.

Divisibility by 9 :

The multiples of 9 are 9, 18, 27, 36, 45, 54,... There are other numbers like 4608, 5283 that are also divisible by 9.

Do you find any pattern when the digits of these numbers are added?
1 + 8 = 9, 2 + 7 = 9, 3 + 6 = 9, 4 + 5 = 9
4 + 6 + 0 + 8 = 18, 5 + 2 + 8 + 3 = 18

All these sums are also divisible by 9.
Is the number 758 divisible by 9?
No. The sum of its digits 7 + 5 + 8 = 20 is also not divisible by 9.
These observations lead us to say that if the sum of the digits of a number is divisible by 9, then the number itself is divisible by 9.

Divisibility by 11 :
The numbers 308, 1331 and 61809 are all divisible by 11.
We form a table and see if the digits in these numbers lead us to some pattern.

We observe that in each case the difference is either 0 or divisible by 11. All these numbers are also divisible by 11.

For the number 5081, the difference of the digits is (5+8) − (1+0) = 12 which is not divisible by 11. The number 5081 is also not divisible by 11.

Thus, to check the divisibility of a number by 11, the rule is, find the difference between the sum of the digits at odd places (from the right) and the sum of the digits at even places (from the right) of the number. If the difference is either 0 or divisible by 11, then the number is divisible by 11.

EXERCISE 3.3

Q1. Using divisibility tests, determine which of the following numbers are divisible by 2;
by 3; by 4; by 5; by 6; by 8; by 9; by 10 ; by 11 (say, yes or no):

A1.

Q2. Using divisibility tests, determine which of the following numbers are divisible by 4; by 8:
(a) 572
(b) 726352
(c) 5500
(d) 6000
(e) 12159
(f) 14560
(g) 21084
(h) 31795072
(i) 1700
(j) 2150

A 2. Divisible by 4 : (a) , (b), (c), (d), (f), (g), (h), (i)
Divisible by 8 : (b), (d), (f), (h)

Q3. Using divisibility tests, determine which of following numbers are divisible by 6:
(a) 297144
(b) 1258
(c) 4335
(d) 61233
(e) 901352
(f) 438750
(g) 1790184
(h) 12583
(i) 639210
(j) 17852

A3. (a), (f), (g), (i)

Q4. Using divisibility tests, determine which of the following numbers are divisible by 11:
(a) 5445
(b) 10824
(c) 7138965
(d) 70169308
(e) 10000001
(f) 901153

A4. (a), (b), (d), (e), (f)

Q5. Write the smallest digit and the greatest digit in the blank space of each of the following numbers so that the number formed is divisible by 3 :
(a) __ 6724
(b) 4765 __ 2

A 5. (a) 2 and 8
(b) 0 and 9

Q6. Write a digit in the blank space of each of the following numbers so that the number formed is divisible by 11 :
(a) 92 __ 389
(b) 8 __ 9484

A 6. (a) 8
(b) 6