Math CentralQuandaries & Queries


Question from debbie, a parent:

hi, i have a daughter and she asked me a maths question I cannot solve. I was just wondering if you can give me the answers plus the working out so I could explain to my daughter,

1. The leftmost digit of a six-digit number N is 1. If this digit is removed and then written as a rightmost digit, the number thus obtained is three times N .Find N.

2. Four friends are racing together down a flight of stairs. A goes 2 steps at a time, B 3 steps at a time. C 4 steps at a time and D 5 steps at a time. The only steps which all four tread on are the top one and the bottom one. How many stairs in the flight were stepped on exactly once?

Hi Debbie,

I'll give you some help getting started and suggest that you show your daughter my suggestions and let her work on her own to attempt to solve the problems.

Problem 1:

N = 1 _ _ _ _ _ and 3 × N = _ _ _ _ _ 1 where the blanks are digits in the same order in N as in 3 × N. Written using the traditional method we use for multiplication this is

1 _ _ _ _  
_ _ _ _   1

I want to focus on the digit that goes in the pink box. Three times that digit has a units digit of 1 so the digit 7 must go in the pink box on the first line and hence also in the pink box in the second line. Thus we have

1 _ _ _   7
_ _ _   7 1

Again, what goes in the pink box? Performing the multiplication, 3 time 7 is 21 so you put a 1 in the bottom row and carry the 2. three times the digit in the pink box in the first row must have units digit 5 so that adding the carry of 2 places the 7 in the bottom row. three times 5 is 15 so this time the digit 5 goes in both pink boxes.

Continue this process.

Problem 2:

Think about numbering the first step 1, the second 2 and so on. A steps on steps that are divisible by 2, B steps on steps that are divisible by 3, C steps on steps that are divisible by 4 and D steps on steps that are divisible by 5. Thus the bottom step is divisible by 2, 3, 4, and 5. What is the smallest positive integer that is divisible by 2, 3, 4 and 5. (It's 60 but have your daughter determine this herself.)

My approach to this problem is very "hands on". Neatly write the integers from 1 to 60 maybe in a table wit rows 1 to 10 then 11 to 20 then 21 to 30 and so on. Get 4 different coloured pencils or markers. With the first pencil cross out all the multiples of 2, with the second cross out all the multiples of 3. Do the same with the remaining two pencils and multiples of 4 and 5.

What numbers only got crossed out once? What is the connection to 2, 3, 4 and 5?

I hope this helps,


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