example-1Multiple choice
Generalised form of a three-digit number xyz is
- (a)x + y + z
- (b)100x + 10y + z
- (c)100z + 10y + x
- (d)100y + 10x + z
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Solution : The correct answer is (b).
Chapter 13
Class 8 Mathematics · 52 questions · 52 with answers
Solved examples
example-2Multiple choice
The usual form of 100a + b + 10c is
- (a)abc
- (b)cab
- (c)bac
- (d)acb
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Solution : The correct answer is (d).
example-3Multiple choice
If 5 × A = CA then the values of A and C are
- (a)A = 5, C = 1
- (b)A = 4, C = 2
- (c)A = 5, C = 2
- (d)A = 2, C = 5
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Solution : The correct answer is (c).
example-4Multiple choice
If 5 A + 25 is equal to B 2, then the value of A + B is
- (a)15
- (b)10
- (c)8
- (d)7
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Solution : The correct answer is (a). In examples 5 to 7, fill in the blanks to make the statements true.
example-5Fill in the blanks
The number ab – ba where a and b are digits and a > b is divisible by ________.
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Solution : 9.
example-6Fill in the blanks
When written in usual form 100a + 10c + 9 is equal to ________.
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Solution : ac 9
example-7Fill in the blanks
If AB × B = 9B, then A = _________, B = _________.
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Solution : 9, 1 In examples 8 to 10, state whether the statements are true (T) or false (F).
example-8Short answer
If abc, cab, bca are three digit numbers formed by the digits a, b, and c then the sum of these numbers is always divisible by 37.
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Solution : True.
example-9Short answer
Let ab be a two-digit number, then ab + ba is divisible by 9.
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Solution : False.
example-10Short answer
If a number is divisible by 2 and 4, then it will be divisible by 8.
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Solution : False.
example-11Short answer
A three-digit number 42x is divisible by 9. Find the value of x.
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Solution : Since 42x is divisible by 9, the sum of its digits, i.e. 4 + 2 + x must be divisible by 9. i.e. 6 + x is divisible by 9 i.e. 6 + x = 9 or 18, _____. Since x is a digit, therefore 6 + x = 9 or, x = 3.
example-12Short answer
Find the value of A and B if 41 A +B 4
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Solution : From ones column A + 4 gives a number whose ones digit is 2. So, A = 8. The value of B can be obtained by solving 2 + B is a number whose ones digit is 1. So, B = 9. 418 + 94
example-13Short answer
Suppose that the division x ÷ 5 leaves a remainder 4 and the division x ÷ 2 leaves a remainder 1. Find the ones digit of x.
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Solution : Since x ÷ 5 leaves a remainder 4, so ones digit of x can be 4 or 9. Also, since x ÷ 2 leaves a remainder 1, so ones digit must be 9 only.
example-14Long answer
If 756x x is divisible by 11, where x is a digit find the value of x . Understand and Explore the problem • What is given in the question? A four digit number 756x is divisible by 11. • Which property is required to solve the problem? Divisibility of a number by 11. Plan a Strategy • Find the sum of the digits of given number 756x at odd places. • Find the sum of the digits of 756x at even places. • Find the difference of step 1 and step 2. Solve • Given y = 2x • Sum of digits at odd places = x + 5 • Sum of digits at even places = 6 + 7 = 13 • Difference = (x + 5) – 13 =x–8 Now (x – 8) should be equal to 0 or a multiple of 11 (i.e. 11, 22, 33, ..., etc.) x–8=0 x = 8 or x – 8 = 11 x = 11 + 8 = 19 • Since x is a digit so it can take values from 0 – 9 Hence x = 8 Required number is 7568. Revise • 7568 Sum of digits at odd places = 5 + 8 = 13 Sum of digits at even places = 6 + 7 = 13 Difference = 13 – 13 = 0 So Value of x is correct.
Questions
Q1Multiple choice
What would be the value of y, if 277y is divisible by 11? In each of the questions 1 to 17, out of the four options, only one is correct. Write the correct answer. 1. Generalised form of a four-digit number abdc is
- (a)1000 a + 100 b + 10 c + d
- (b)1000 a + 100 c + 10 b + d
- (c)1000 a + 100 b + 10 d + c
- (d)a × b × c × d
Show answer
(c) 1000 a + 100 b + 10 d + c
Q2Multiple choice
Generalised form of a two-digit number xy is
- (a)x + y
- (b)10x + y
- (c)10x – y
- (d)10y + x
Show answer
(b) 10x + y
Q3Multiple choice
The usual form of 1000a + 10b + c is
- (a)abc
- (b)abco
- (c)aobc
- (d)aboc
Show answer
(c) aobc
Q4Multiple choice
Let abc be a three-digit number. Then abc – cba is not divisible by
- (a)9
- (b)11
- (c)18
- (d)33
Show answer
(c) 18
Q5Multiple choice
The sum of all the numbers formed by the digits x, y and z of the number xyz is divisible by
- (a)11
- (b)33
- (c)37
- (d)74
Show answer
(c) 37
Q6Multiple choice
A four-digit number aabb is divisible by 55. Then possible value(s) of b is/are
- (a)0 and 2
- (b)2 and 5
- (c)0 and 5
- (d)7
Show answer
(c) 0 and 5
Q7Multiple choice
Let abc be a three digit number. Then abc + bca + cab is not divisible
- (a)a + b + c
- (b)3
- (c)37
- (d)9
Show answer
(d) 9
Q8Multiple choice
A four-digit number 4ab5 is divisible by 55. Then the value of b – a is
- (a)0
- (b)1
- (c)4
- (d)5
Show answer
(b) 1
Q9Multiple choice
If abc is a three digit number, then the number abc – a – b – c is divisible by
- (a)9
- (b)90
- (c)10
- (d)11
Show answer
(a) 9
Q10Multiple choice
A six-digit number is formed by repeating a three-digit number. For example 256256, 678678, etc. Any number of this form is divisible
- (a)7 only
- (b)11 only
- (c)13 only
- (d)1001
Show answer
(d) 1001
Q11Multiple choice
If the sum of digits of a number is divisible by three, then the number is always divisible by
- (a)2
- (b)3
- (c)6
- (d)9
Show answer
(b) 3
Q12Multiple choice
If x + y + z = 6 and z is an odd digit, then the three-digit number xyz is
- (a)an odd multiple of 3
- (b)odd multiple of 6
- (c)even multiple of 3
- (d)even multiple of 9
Show answer
(a) an odd multiple of 3
Q13Multiple choice
If 5 A + B 3 = 65, then the value of A and B is
- (a)A = 2, B = 3
- (b)A = 3, B = 2
- (c)A = 2, B = 1
- (d)A = 1, B = 2
Show answer
(c) A = 2, B = 1
Q14Multiple choice
If A 3 + 8 B = 150, then the value of A + B is
- (a)13
- (b)12
- (c)17
- (d)15
Show answer
(a) 13
Q15Multiple choice
If 5 A × A = 399, then the value of A is
- (a)3
- (b)6
- (c)7
- (d)9
Show answer
(c) 7
Q16Multiple choice
If 6 A × B = A 8 B, then the value of A – B is
- (a)–2
- (b)2
- (c)–3
- (d)3
Show answer
(a) –2
Q17Multiple choice
Which of the following numbers is divisible by 99
- (a)913462
- (b)114345
- (c)135792
- (d)3572406 In questions 18 to 33, fill in the blanks to make the statements true.
Show answer
(b) 114345
Q18Fill in the blanks
3134673 is divisible by 3 and ______.
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9
Q19Fill in the blanks
20x3 is a multiple of 3 if the digit x is ______ or ______ or ______.
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1, 4, 7
Q20Fill in the blanks
3x5 is divisible by 9 if the digit x is __________.
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1
Q21Fill in the blanks
The sum of a two–digit number and the number obtained by reversing the digits is always divisible by __________.
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11
Q22Fill in the blanks
The difference of a two–digit number and the number obtained by reversing its digits is always divisible by ___________.
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9
Q23Fill in the blanks
The difference of three-digit number and the number obtained by putting the digits in reverse order is always divisible by 9 and ___________. 2 B
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11
Q24Fill in the blanks
If + A B then A = ______ and B = ______. 8 A A B
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A = 6, B = 3
Q25Fill in the blanks
If × B then A = ______ and B = ______. 9 6 B 1
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A = 2, B = 4 (four)
Q26Fill in the blanks
If × B then B = _______. 4 9B
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B = 7
Q27Fill in the blanks
1 x 35 is divisible by 9 if x = _______.
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x = 0
Q28Fill in the blanks
A four-digit number abcd is divisible by 11, if d + b = _______ or _____
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a + c or 12 (a + c)
Q29Fill in the blanks
A number is divisible by 11 if the differences between the sum of digits at its odd places and that of digits at the even places is either 0 or divisible by ______.
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11
Q30Fill in the blanks
If a 3-digit number abc is divisible by 11, then ______ is either 0 or mul tiple of 11.
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(a + c) – b
Q31Fill in the blanks
If A × 3 = 1A, then A = ______.
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5
Q32Fill in the blanks
If B × B = AB, then either A = 2, B = 5 or A = ______, B = ______.
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values, A = 3, B = 6
Q33Fill in the blanks
If the digit 1 is placed after a 2-digit number whose tens is t and ones digit is u, the new number is ______. State whether the statements given in questions 34 to 44 are true (T) or false (F):
Show answer
t 41
Q34Short answer
A two-digit number ab is always divisible by 2 if b is an even number.
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True
Q35Short answer
A three-digit number abc is divisible by 5 if c is an even number.
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False
Q36Short answer
A four-digit number abcd is divisible by 4 if ab is divisible by 4.
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False
Q37Short answer
A three-digit number abc is divisible by 6 if c is an even number and a + b + c is a multiple of 3.
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True
Q38Short answer
Number of the form 3N + 2 will leave remainder 2 when divided by 3.
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True
Q39Short answer
Number 7N + 1 will leave remainder 1 when divided by 7.
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True
Q40Short answer
If a number a is divisible by b, then it must be divisible by each factor of b.
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True
Q41Short answer
If AB × 4 = 192, then A + B = 7.
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False
Q42Short answer
If AB + 7C = 102, where B ≠ 0, C ≠ 0, then A + B + C = 14.
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True
Q43Short answer
If 213x 27 is divisible by 9, then the value of x is 0.
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False
Q44Short answer
If N ÷ 5 leaves remainder 3 and N ÷ 2 leaves remainder 0, then N ÷ 10 leaves remainder 4. Solve the following :
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False
Q45Short answer
Find the least value that must be given to number a so that the number 91876a2 is divisible by 8. 1 P
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a = 3
Q46Short answer
If × P where Q – P = 3, then find the values of P and Q. Q 6
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P = 6 and Q = 9
Q47Short answer
If 1AB + CCA = 697 and there is no carry–over in addition, find the value of A + B + C.
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12
Q48Short answer
A five-digit number AABAA is divisible by 33. Write all the numbers of this form.
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33033, 66066, 99099
Q49Short answer
Find the value of the letters in each of the following questions. A A +A A XA Z
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A = 9, Z = 8, X = 1
Q50Short answer
85 51. B6 52. 1 BA +4 A +8 A + A BA BC3 CA2 8 A 2
Show answer
A = 8, B = 1, C = 3
Q53Short answer
C BA 54. B AA 55. A 01 B +C BA +B AA +1 0 A B 1 A 30 3 A 8 B 10 8
Show answer
A = 5, B = 6, C = 7
Q56Multiple choice
AB 57. AB 58. AA × 6 × AB × A C68 6 A B C A B and B – A = 1 59. AB 60. 8ABC 61. If 2A7 ÷ A = 33, – B 7 –ABC5 then find the 4 5 D488 value of A. 62. 212 x 5 is a multiple of 3 and 11. Find the value of x. 63. Find the value of k where 31k 2 is divisible by 6. 64. 1y3y6 is divisible by 11. Find the value of y. 65. 756 x is a multiple of 11, find the value of x. 66. A three-digit number 2 a 3 is added to the number 326 to give a three-digit number 5b9 which is divisible by 9. Find the value of b –a. 67. Let E = 3, B = 7 and A = 4. Find the other digits in the sum BASE +B A L L G AM E S 68. Let D = 3, L = 7 and A = 8. Find the other digits in the sum + AS + A BULL 69. If from a two-digit number, we subtract the number formed by reversing its digits then the result so obtained is a perfect cube. How many such numbers are possible? Write all of them. 70. Work out the following multiplication. 12345679 × 9 Use the result to answer the following questions.
- (a)What will be 12345679 × 45?
- (b)What will be 12345679 × 63?
- (c)By what number should 12345679 be multiplied to get 888888888?
- (d)By what number should 12345679 be multiplied to get 999999999? 71. Find the value of the letters in each of the following: (i) PQ (ii) 2 L M × 6 + LM1 QQQ M18 72. If 148101B095 is divisible by 33, find the value of B. 73. If 123123A4 is divisible by 11, find the value of A. 74. If 56x32y is divisible by 18, find the least value of y. 1. Polygonal Numbers Study the patterns given below and extend it. We already know about square numbers. * * * * * * * * * * * * * * 1 4 9 Draw two more. Here for the first square number, use 12; for the second square number, use 22. To find the third square number use 32 and so on. Write the nth square number. Now let’s move to triangular numbers. * * * * * * * * * * * * * * * * * * * * 1 3 6 10 Find the next triangular number. n × (n + 1) To find the nth triangular number we use the formula Are you familiar with pentagonal numbers? First three are given to you. Write the next one * * * * * * 1 * * * * * * * * * 5 * * * Draw the dot patterns for the next pentagonal number. Count the number of dots inside the entire shape and write the number under the shape. 2. Put tick mark in the appropriate boxes if the given numbers are divisible by any of 2, 3, 4, 5, 6, 8, 10, 11 numbers. S.No. Number Divisible by 2 3 4 5 6 7 8 9 10 11 1. 40185 2. 92286 3. 56390 4. 419562 5. 10593248 3. Cross Number Puzzle Fill in the blank spaces in the cross number puzzle using following clues. Down (a) 59 ______ 63 ÷ 33 (b) 81 ______ 42 ÷ 6 (c) 7 ______ 6988 ÷ 11 (d) 37604 ______ 5 ÷ 15 (e) 56 ______ ÷ 10 Across (f) 90 ______ 815 ÷ 15 (g) 3514 ______ ÷ 12 (h) 4 ______ 07 ÷ 7 (i) 8 ______ 558 ÷ 6 (j) 6 ______ 5 ÷ 55 MATHEMATICS Rough Work UNIT-13 Rough Work PLAYING WITH NUMBERS 419 MATHEMATICS Rough Work
Show answer
(a) What will be 12345679 × 45?
(b) What will be 12345679 × 63?
(c) By what number should 12345679 be multiplied to get 888888888?