Playing with Numbers — Exercise 4(B)

Test the divisibility of each of the following numbers by 2:

(i) 96

(ii) 830

(iii) 367

(iv) 922

(v) 881

(vi) 934

(vii) 1089

(viii) 13058

Answer

According to the divisibility test of 2, a given number is divisible by 2 only when its units digit is even, i.e. 0, 2, 4, 6 or 8.

(i) 96 — Units digit is 6, which is even.

Hence, 96 is divisible by 2.

(ii) 830 — Units digit is 0, which is even.

Hence, 830 is divisible by 2.

(iii) 367 — Units digit is 7, which is not even.

Hence, 367 is not divisible by 2.

(iv) 922 — Units digit is 2, which is even.

Hence, 922 is divisible by 2.

(v) 881 — Units digit is 1, which is not even.

Hence, 881 is not divisible by 2.

(vi) 934 — Units digit is 4, which is even.

Hence, 934 is divisible by 2.

(vii) 1089 — Units digit is 9, which is not even.

Hence, 1089 is not divisible by 2.

(viii) 13058 — Units digit is 8, which is even.

Hence, 13058 is divisible by 2.

Test the divisibility of each of the following numbers by 3:

(i) 258

(ii) 774

(iii) 584

(iv) 1033

(v) 2571

(vi) 6654

(vii) 10385

(viii) 35742

Answer

According to the divisibility test of 3, a given number is divisible by 3 only when the sum of its digits is divisible by 3.

(i) 258 — Sum of digits = 2 + 5 + 8 = 15, which is divisible by 3.

Hence, 258 is divisible by 3.

(ii) 774 — Sum of digits = 7 + 7 + 4 = 18, which is divisible by 3.

Hence, 774 is divisible by 3.

(iii) 584 — Sum of digits = 5 + 8 + 4 = 17, which is not divisible by 3.

Hence, 584 is not divisible by 3.

(iv) 1033 — Sum of digits = 1 + 0 + 3 + 3 = 7, which is not divisible by 3.

Hence, 1033 is not divisible by 3.

(v) 2571 — Sum of digits = 2 + 5 + 7 + 1 = 15, which is divisible by 3.

Hence, 2571 is divisible by 3.

(vi) 6654 — Sum of digits = 6 + 6 + 5 + 4 = 21, which is divisible by 3.

Hence, 6654 is divisible by 3.

(vii) 10385 — Sum of digits = 1 + 0 + 3 + 8 + 5 = 17, which is not divisible by 3.

Hence, 10385 is not divisible by 3.

(viii) 35742 — Sum of digits = 3 + 5 + 7 + 4 + 2 = 21, which is divisible by 3.

Hence, 35742 is divisible by 3.

Test the divisibility of each of the following numbers by 4:

(i) 562

(ii) 8176

(iii) 56318

(iv) 35344

(v) 909032

(vi) 77752

Answer

According to the divisibility test of 4, a given number is divisible by 4 only when the number formed by its last two digits is divisible by 4.

(i) 562 — Number formed by the last two digits = 62, which is not divisible by 4.

Hence, 562 is not divisible by 4.

(ii) 8176 — Number formed by the last two digits = 76 = 4 × 19, which is divisible by 4.

Hence, 8176 is divisible by 4.

(iii) 56318 — Number formed by the last two digits = 18, which is not divisible by 4.

Hence, 56318 is not divisible by 4.

(iv) 35344 — Number formed by the last two digits = 44 = 4 × 11, which is divisible by 4.

Hence, 35344 is divisible by 4.

(v) 909032 — Number formed by the last two digits = 32 = 4 × 8, which is divisible by 4.

Hence, 909032 is divisible by 4.

(vi) 77752 — Number formed by the last two digits = 52 = 4 × 13, which is divisible by 4.

Hence, 77752 is divisible by 4.

Test the divisibility of each of the following numbers by 5:

(i) 1095

(ii) 2746

(iii) 8185

(iv) 4040

(v) 98762

(vi) 77770

Answer

According to the divisibility test of 5, a given number is divisible by 5 only when its units digit is 0 or 5.

(i) 1095 — Units digit is 5.

Hence, 1095 is divisible by 5.

(ii) 2746 — Units digit is 6, which is neither 0 nor 5.

Hence, 2746 is not divisible by 5.

(iii) 8185 — Units digit is 5.

Hence, 8185 is divisible by 5.

(iv) 4040 — Units digit is 0.

Hence, 4040 is divisible by 5.

(v) 98762 — Units digit is 2, which is neither 0 nor 5.

Hence, 98762 is not divisible by 5.

(vi) 77770 — Units digit is 0.

Hence, 77770 is divisible by 5.

Test the divisibility of each of the following numbers by 8:

(i) 7304

(ii) 8612

(iii) 44440

(iv) 265472

Answer

According to the divisibility test of 8, a given number is divisible by 8 only when the number formed by its last three digits is divisible by 8.

(i) 7304 — Number formed by the last three digits = 304 = 8 × 38, which is divisible by 8.

Hence, 7304 is divisible by 8.

(ii) 8612 — Number formed by the last three digits = 612, which is not divisible by 8.

Hence, 8612 is not divisible by 8.

(iii) 44440 — Number formed by the last three digits = 440 = 8 × 55, which is divisible by 8.

Hence, 44440 is divisible by 8.

(iv) 265472 — Number formed by the last three digits = 472 = 8 × 59, which is divisible by 8.

Hence, 265472 is divisible by 8.

Test the divisibility of each of the following numbers by 10:

(i) 4005

(ii) 30303

(iii) 64610

(iv) 85730

Answer

According to the divisibility test of 10, a given number is divisible by 10 only when its units digit is 0.

(i) 4005 — Units digit is 5, which is not 0.

Hence, 4005 is not divisible by 10.

(ii) 30303 — Units digit is 3, which is not 0.

Hence, 30303 is not divisible by 10.

(iii) 64610 — Units digit is 0.

Hence, 64610 is divisible by 10.

(iv) 85730 — Units digit is 0.

Hence, 85730 is divisible by 10.

Test the divisibility of each of the following numbers by 11:

(i) 1452

(ii) 9416

(iii) 818532

(iv) 84524

Answer

According to the divisibility test of 11, a number is divisible by 11 if the difference between the sum of its digits at odd places and the sum of its digits at even places is either 0 or a number divisible by 11.

(i) The given number is 1452.

Sum of its digits at odd places = (2 + 4) = 6.

Sum of its digits at even places = (5 + 1) = 6.

Difference of the above sums = (6 - 6) = 0.

Hence, 1452 is divisible by 11.

(ii) The given number is 9416.

Sum of its digits at odd places = (6 + 4) = 10.

Sum of its digits at even places = (1 + 9) = 10.

Difference of the above sums = (10 - 10) = 0.

Hence, 9416 is divisible by 11.

(iii) The given number is 818532.

Sum of its digits at odd places = (2 + 5 + 1) = 8.

Sum of its digits at even places = (3 + 8 + 8) = 19.

Difference of the above sums = (19 - 8) = 11, which is divisible by 11.

Hence, 818532 is divisible by 11.

(iv) The given number is 84524.

Sum of its digits at odd places = (4 + 5 + 8) = 17.

Sum of its digits at even places = (2 + 4) = 6.

Difference of the above sums = (17 - 6) = 11, which is divisible by 11.

Hence, 84524 is divisible by 11.

Find all possible values of $x$ for which the number $534x2$ is divisible by 3. Also, find each such number.

Answer

The number $534x2$ is divisible by 3.

⇒ Sum of its digits = $(5 + 3 + 4 + x + 2) = (14 + x)$ must be divisible by 3.

⇒ $14 + x$ = 15, 18, 21, 24, ...............

⇒ $x$ = 15 - 14, 18 - 14, 21 - 14, 24 - 14, ........

⇒ $x$ = 1, 4, 7, 10, ...............

Since $x$ is a digit, it cannot be more than 9.

Hence, the possible values of $x$ are $x = 1 \text{ or } x = 4 \text{ or } x = 7$.

Hence, all the required numbers are 53412, 53442 and 53472.

Find the least number which should replace * in the number 89*2 to make it exactly divisible by:

(i) 3

(ii) 9

Answer

(i) The number 89*2 is divisible by 3.

⇒ Sum of its digits = (8 + 9 + * + 2) = (19 + *) must be divisible by 3.

⇒ 19 + * = 21, 24, 27, ...............

⇒ * = 21 - 19, 24 - 19, 27 - 19, ........

⇒ * = 2, 5, 8, ...............

Since * is a digit and the least possible value is required, we take ∗ = 2.

Hence, * is to be replaced by 2.

(ii) The number 89*2 is divisible by 9.

⇒ Sum of its digits = (8 + 9 + * + 2) = (19 + *) must be divisible by 9.

⇒ 19 + * = 27, 36, ...............

⇒ * = 27 - 19, 36 - 19, ........

⇒ * = 8, 17, ...............

Since * is a digit and the least possible value is required, we take * = 8.

Hence, * is to be replaced by 8.

Find the value of $z$ for which the number $417z8$ is divisible by 9. Also, find the number.

Answer

The number $417z8$ is divisible by 9.

⇒ Sum of its digits = $(4 + 1 + 7 + z + 8) = (20 + z)$ must be divisible by 9.

⇒ $20 + z$ = 27, 36, ...............

⇒ $z$ = 27 - 20, 36 - 20, ........

⇒ $z$ = 7, 16, ...............

Since $z$ is a digit, it cannot be more than 9.

∴ $z = 7$

∴ The required number = 41778

Hence, $\bm{z = 7}$ and the required number is 41778.

Give five examples of numbers, each one of which is divisible by 3 but not divisible by 9.

Answer

A number is divisible by 3 only when the sum of its digits is divisible by 3, and it is divisible by 9 only when the sum of its digits is divisible by 9.

So, we choose numbers whose digit-sum is divisible by 3 but not by 9.

12 — Sum of digits = 1 + 2 = 3, which is divisible by 3 but not by 9.

15 — Sum of digits = 1 + 5 = 6, which is divisible by 3 but not by 9.

21 — Sum of digits = 2 + 1 = 3, which is divisible by 3 but not by 9.

24 — Sum of digits = 2 + 4 = 6, which is divisible by 3 but not by 9.

33 — Sum of digits = 3 + 3 = 6, which is divisible by 3 but not by 9.

Hence, 12, 15, 21, 24 and 33 are five numbers each of which is divisible by 3 but not divisible by 9.

Give five examples of numbers, each one of which is divisible by 4 but not divisible by 8.

Answer

A number is divisible by 4 only when the number formed by its last two digits is divisible by 4, and it is divisible by 8 only when the number formed by its last three digits is divisible by 8.

12 = 4 × 3, so 12 is divisible by 4, but 12 is not divisible by 8.

20 = 4 × 5, so 20 is divisible by 4, but 20 is not divisible by 8.

28 = 4 × 7, so 28 is divisible by 4, but 28 is not divisible by 8.

36 = 4 × 9, so 36 is divisible by 4, but 36 is not divisible by 8.

44 = 4 × 11, so 44 is divisible by 4, but 44 is not divisible by 8.

Hence, 12, 20, 28, 36 and 44 are five numbers each of which is divisible by 4 but not divisible by 8.