Page 52 - Start Up Mathematics_6

Example 4: What is the sum of (a) two odd numbers, (b) two even numbers and (c) one odd and
one even number?

Solution: (a) Sum of two odd numbers is an even number. Do you know?
(b) Sum of two even numbers is again an even number. The expression
2
(c) Sum of an odd and even number is an odd number. n – n + 41, where n
is a whole number
Example 5: The numbers 17 and 71 are prime numbers. Both these from 0 to 40, results
numbers have the same digits 1 and 7. Find other such pairs in a prime number.
of prime numbers up to 100.

Solution: Such pairs are:
(a) 13 and 31 (b) 37 and 73 (c) 79 and 97.
Example 6: Write down separately the prime and composite numbers less than 15. Find their
sums. Which is greater?

Solution: Prime numbers less than 15 are 2, 3, 5, 7, 11 and 13.
Composite numbers less than 15 are 4, 6, 8, 9, 10, 12 and 14.
Sum of prime numbers less than 15 = 2 + 3 + 5 + 7 + 11 + 13 = 41

Sum of composite numbers less than 15 = 4 + 6 + 8 + 9 + 10 + 12 + 14 = 63
Hence, the sum of composite numbers is greater.

Example 7: Express the following as the sum of two odd primes.
(a) 44 (b) 72 (c) 24 (d) 26
Solution: (a) 44 = 13 + 31 (b) 72 = 31 + 41 (c) 24 = 11 + 13 (d) 26 = 3 + 23

Example 8: Is there any twin prime between 70 and 80?
Solution: Yes, the pair 71 and 73 is a twin prime between 70 and 80.

Example 9: Write seven consecutive composite numbers less than 100 with no prime number
between them. What is the maximum number of consecutive prime numbers with
no composite number between them?
Solution: Seven consecutive composite numbers less than 100 with no prime number between
them are 90, 91, 92, 93, 94, 95 and 96. The only pair of consecutive prime numbers
is 2 and 3.
Example 10: Express each of the following numbers as the sum of three odd prime numbers:

(a) 91 (b) 15 (c) 61
Solution: (a) 91 = 5 + 19 + 67 (b) 15 = 3 + 5 + 7 (c) 61 = 13 + 17 + 31

Example 11: Examine whether the given number is divisible by any prime number less than or
equal to the square root of the given number. If yes, then find whether it is a composite
number or a prime number.
(a) 403 (b) 617
Solution: (a) Since square root of 403 lies between 20 and 21.
We check whether 403 is divisible by any of the prime numbers 2, 3, 5, 7, 11, 13,
17 and 19. By trial, we find that 403 is divisible by 13. Hence it is a composite
number.

44