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International Contest-Game Math Worksheet With Answers - Grade 5 And 6, Kangaroo, 2007 Page 7

Solution: Let Anna’s number has a tens and b units, i.e., it is written by the digits a, b, in this

order. Let us denote this number by ab . Therefore, Ben’s number will be written by the

sequence of digits a, b, a, b (we will denote it by abab ). We can represent the number abab as

follows:

abab

100

101

. In the last step, we used the

distributive property of multiplication. Therefore, when Ben’s number abab is divided by

Anna’s number ab the result will be 101.

Answer: B.

23. Five integer numbers are written around a circle in a way that no two or three adjacent

numbers have a sum divisible by 3. How many of these five numbers are divisible by 3?

A) 0

B) 1

C) 2

D) 3

E) Impossible to determine

Solution: The condition that the sum of any two adjacent numbers is not divisible by 3

translates into the fact that the remainders of these numbers when divided by 3 cannot be a

sequence of 1 and 2, and cannot be two zeroes. So, two adjacent numbers can only have

remainders (0 ,1), (1, 0), (0, 2), (2, 0), (1, 1) or (2, 2). On the other hand, the sum of any three

adjacent

numbers

also

not

divisible

therefore,

the

combinations

(0, 0, 0), (1, 1, 1), (1, 0, 2), or (2, 0, 1) are not possible.

Let us assume that there is no multiple of 3 among the five numbers. It follows that all

five remainders are either 1 or 2, and evidently, they cannot be arranged to comply with the

above restrictions.

Let us assume that there is only one multiple of 3. Then, the only possibilities for the

numbers adjacent to it, are either 1 and 1 or 2 and 2. Without loss of generality, let us assume

these numbers are 1 and 1. So, we already have a sequence of 1, 0, 1 and we must add two more

numbers to the right of it. These numbers cannot be (1,1), or (2, 1) or (1, 2) or (2, 2) or (0, 2) or

(2, 0). The possibilities are (1, 0) or (0, 1) only. Therefore, we will need two multiples of 3.

If we have three or more multiples of 3 among these five numbers, there always will be

three of them adjacent to each other, which is not allowed.

Answer: C

…………………

International Contest-Game Math Worksheet With Answers - Grade 5 And 6, Kangaroo, 2007 Page 7