Class six math chapter 1- Exercise 1.2 solution
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Exercise 1.2 solution
1. Write the prime numbers between 30 and 70.
Solution:
The prime numbers between 30 and 70 are: 31, 37, 41, 43, 47, 53, 59, 61, 67
2. Determine the co-prime pairs:
(a) 27, 54
Here,
27 = 1 × 3 × 3 × 3
54 = 1 × 2 × 3 × 3 × 3
The divisors of 27 are 1, 3, 9, 27
The divisors of 54 are 1, 2, 3, 6, 9, 18, 27, 54
Common divisors of 27 and 54 are: 1, 3, 9, 27
Therefore, 27 and 54 are not co-prime.
(b) 63, 91
Here,
63 = 1 × 3 × 3 × 7
91 = 1 × 7 × 13
The divisors of 63 are 1, 3, 7, 9, 21, 63
The divisors of 91 are 1, 7, 13, 91
Since 63 and 91 have a common divisor of 7 other than 1, they are not co-prime.
(c) 189, 210
Here,
189 = 1 × 3 × 3 × 3 × 7
210 = 1 × 2 × 3 × 5 × 7
The divisors of 189 are 1, 3, 7, 9, 21, 27, 63
The divisors of 210 are 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210
Common divisors of 189 and 210 are 1, 3, 7, 21.
Therefore, 189 and 210 are not co-prime.
(d) 52, 97
Here,
52 = 1 × 2 × 2 × 13
97 = 1 × 97
The divisors of 52 are 1, 2, 4, 13, 26, 52
The divisors of 97 are 1, 97
Since the only common divisor of 52 and 97 is 1, they are co-prime.
3. Determine which of the following numbers are divisible by the indicated numbers:
(a) Divisible by 3: 545, 6774, 8535
Solution:
For 545:
The sum of the digits is 5 + 4 + 5 = 14, which is not divisible by 3.
Therefore, 545 is not divisible by 3.
For 6774:
The sum of the digits is 6 + 7 + 7 + 4 = 24, which is divisible by 3.
Therefore, 6774 is divisible by 3.
For 8535:
The sum of the digits is 8 + 5 + 3 + 5 = 21, which is divisible by 3.
Therefore, 8535 is divisible by 3.
(b) Divisible by 4: 8542, 2184, 5274
Solution:
For 8542:
The number formed by the last two digits is 42, which is not divisible by 4.
Therefore, 8542 is not divisible by 4.
For 2184:
The number formed by the last two digits is 84, which is divisible by 4.
Therefore, 2184 is divisible by 4.
For 5274:
The number formed by the last two digits is 74, which is not divisible by 4.
Therefore, 5274 is not divisible by 4.
(c) Divisible by 6: 2184, 1074, 7832
Solution:
For 2184:
The last digit is 4, which is divisible by 2. The sum of the digits is 2 + 1 + 8 + 4 = 15, which is divisible by 3.
Therefore, 2184 is divisible by 6.
For 1074:
The last digit is 4, which is divisible by 2. The sum of the digits is 1 + 0 + 7 + 4 = 12, which is divisible by 3.
Therefore, 1074 is divisible by 6.
For 7832:
The last digit is 2, which is divisible by 2. The sum of the digits is 7 + 8 + 3 + 2 = 20, which is not divisible by 3.
Therefore, 7832 is not divisible by 6.
(d) Divisible by 9: 5075, 1737, 2193
Solution:
For 5075:
The sum of the digits is 5 + 0 + 7 + 5 = 17, which is not divisible by 9.
Therefore, 5075 is not divisible by 9.
For 1737:
The sum of the digits is 1 + 7 + 3 + 7 = 18, which is divisible by 9.
Therefore, 1737 is divisible by 9.
For 2193:
The sum of the digits is 2 + 1 + 9 + 3 = 15, which is not divisible by 9.
Therefore, 2193 is not divisible by 9.
4. Which digits should be placed in the blanks to make the number divisible by 9?
(a) 547□3
Solution:
The sum of the digits is 5 + 4 + 7 + □ + 3 = 21 + □. To be divisible by 9, the sum must be 27.
Thus, □ = 6.
So, the missing digit is 6.
(b) 812□74
Solution:
The sum of the digits is 8 + 1 + 2 + □ + 7 + 4 = 22 + □. To be divisible by 9, the sum must be 27.
Thus, □ = 5.
So, the missing digit is 5.
(c) 4157□
Solution:
The sum of the digits is 4 + 1 + 5 + 7 + □ = 17 + □. To be divisible by 9, the sum must be 18.
Thus, □ = 1.
So, the missing digit is 1.
(d) 574□
Solution:
The sum of the digits is 5 + 7 + 4 + □ = 16 + □. To be divisible by 9, the sum must be 18 or 27.
Thus, □ = 2 or 9.
So, the missing digits can be 2 or 9.
5. Find the smallest five-digit number divisible by 3.
Solution:
The smallest five-digit number is 10000.
The sum of the digits is 1 + 0 + 0 + 0 + 0 = 1, which is not divisible by 3.
Next multiple of 3 after 1 is 3.
Thus, the smallest number divisible by 3 is 10000 + 2 = 10002.
6. Find the largest seven-digit number divisible by 6.
Solution:
The largest seven-digit number is 9999999.
The sum of the digits is 9 + 9 + 9 + 9 + 9 + 9 + 9 = 63, which is divisible by 3, but the last digit is not divisible by 2.
Thus, subtract 3 from 9999999 to get 9999996. The sum of the digits of 9999996 is 60, divisible by 3, and the last digit is 6, divisible by 2.
Therefore, 9999996 is divisible by 6.
7. Determine if the largest number formed by the digits 3, 0, 5, 2, 7 is divisible by 4 and 5.
Solution:
The largest number formed by 3, 0, 5, 2, 7 is 75320.
For divisibility by 4, the number formed by the last two digits is 20, which is divisible by 4.
For divisibility by 5, the last digit is 0, so it is divisible by 5.
Therefore, 75320 is divisible by both 4 and 5.
Some problems for more practice
1. Write the prime numbers between 50 and 100.
2. Determine the co-prime pairs:
(a) 36, 72
(b) 75, 90
(c) 108, 144
(d) 35, 65
3. Determine which of the following numbers are divisible by the indicated numbers:
(a) Divisible by 6: 360, 780, 900
(b) Divisible by 9: 8191, 3645, 2781
(c) Divisible by 11: 2310, 3322, 4995
(d) Divisible by 13: 104, 526, 780
4. Which digits should be placed in the blanks to make the number divisible by 11?
(a) 6 □ 8941
(b) 213 □ 64
(c) 5483 □
(d) 876 □
5. Find the smallest six-digit number divisible by 9.
6. Find the largest eight-digit number divisible by 12.
7. Determine if the largest number formed by the digits 2, 4, 6, 8, 9 is divisible by 3 and 7.
8. Check divisibility by 2, 3, 5, 7, 11, 13, and 17:
(a) 45542
(b) 98765
(c) 12345
(d) 75813
(e) 84570
9. Determine which numbers are divisible by 4 and 6:
(a) 360, 720, 960
(b) 1234, 1458, 1676
(c) 8910, 10520, 7344
(d) 5678, 6096, 7088
10. Determine the co-prime pairs:
(a) 8 and 15
(b) 10 and 25
(c) 35 and 80
(d) 95 and 125
(e) 27 and 38
11. Determine which numbers are divisible by both 9 and 11:
(a) 36542, 76543, 89120
(b) 45678, 12345, 98765
(c) 99999, 88888, 77777
(d) 101010, 102020, 103030
12. Find the square root of the following numbers:
(a) 144
(b) 361
(c) 10000
(d) 576
(e) 841
13. Find the smallest six-digit number divisible by both 5 and 9.
14. Find the largest five-digit number divisible by both 7 and 13.
15. Find the largest number divisible by 6 among the following:
(a) 4873
(b) 7524
(c) 6359
(d) 9903
16. Find the factors of the following numbers:
(a) 60
(b) 75
(c) 108
(d) 125
(e) 210
17. Find the GCD (Greatest Common Divisor) and LCM (Least Common Multiple) of the following pairs of numbers:
(a) 24 and 36
(b) 15 and 25
(c) 8 and 12
(d) 50 and 100
(e) 12 and 18
