Number Sense Worksheet With Answer Key - 2008/2009 (Page 4 of 4) in pdf

Number Sense

2008/2009 Circle 2 Problem 5

Solution

a),b) The table below shows the list of remaining (prime) numbers and their digit sums.

No.

2 3 5 7 11 13 17 19 23 29 31 37 41 43

Digit Sum 2 3 5 7

No.

47 53 59 61 67 71 73 79 83 89 97

Digit Sum

Note that:

• the factors of each of the remaining (prime) numbers are just 1 and the number itself;

• each of the numbers 2, 11, 29, 47, and 83 has digit sum 2;

• none of the numbers has digit sum 6.

c)(i) If we consider numbers 1 to 9, only number 6 has a digit sum of 6. We now consider the sum

of the digits of numbers 10 to 99. For these numbers, the sum of the digits varies from 1 (for

number 10) to 18 (for number 99). If this sum is equal to 1, 2, 3, 4, 5, 6, 7, 8 or 9, it is also the

digit sum of the number. Therefore if, initially, the sum of the digits of the number is equal to

6, the digit sum of the number is also equal to 6. If the sum of the digits is initially equal to

10, 11, 12, 13, 14, 15, 16, 17 or 18, then only 15 gives a digit sum of 6.b Therefore if a number

from 1 to 99 has a digit sum of 6, then the sum of its digits must be equal to 6 or to 15.

(ii) Numbers whose digits have sum 6 are 6, 15, 24, 33, 42, 51 and 60. Numbers whose digits have

sum 15 are 69, 78, 87 and 96. Therefore, numbers 6, 15, 24, 33, 42, 51, 60, 69, 78, 87 and 96

have a digit sum of 6.

Extension:

1. Consider the sum of the digits for the numbers 101 to 199, which varies from 2 to 19. As in

part c), it must be equal to 6 or to 15 to get a digit sum of 6. To ﬁnd the numbers whose sum of

digits is equal to 6 or to 15, we can work more rapidly if we only look at the units digit and the

tens digit whose sum must be equal to 5 or to 14, since the hundreds digit is 1. These numbers

are 105, 114, 123, 132, 141, 150, 159, 168, 177, 186 and 195. There are 11 such numbers.

2. We know that a number is divisible by 3 if the sum of its digits is divisible by 3. From part

c), we know that numbers whose digit sum is equal to 6 are such that the sum of their digits

is equal to 6 or to 15. Since these two sums are divisible by 3, the numbers whose digit sum

is equal to 6 must be divisible by 3, and hence will never be prime numbers. Similarly, if the

digit sum of a number is 9, then the number is divisible by 9, and hence cannot be prime.

Number Sense Worksheet With Answer Key - 2008/2009 Page 4