hide

A positive integer hides another number when, by erasing some of its digits, the other number appears. For example, the number 123 hides the numbers 1, 2, 3, 12, 13, and 23, but does not hide 32, 123, and 213.

a) What is the largest three-digit number hidden by 47239?
b) What is the smallest number that hides both 2009 and 9002 simultaneously?

c) Find a multiple of 2009 that hides 2009 and whose units digit is 3.

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hide - Solution

a) To obtain the largest possible three-digit number hidden by 47239, we must first ensure that this number has the largest possible digit in the hundreds place. To do this, we erase the 4 and leave the 7 in the hundreds place. After that, we seek the largest possible digit in the tens place; for this, we erase the 2 and obtain 739, which is the number we are looking for.

b) Since the number we are looking for hides 2009, among its digits are 2, 0, 0, and 9, in this order. Similarly, since it hides 9002, among its digits are 9, 0, 0, and 2 in this order. Therefore, this number has at least six digits: a 2 and a 9 to the left of two 0's and a 2 and a 9 to the right of the same. There are exactly four six-digit numbers of this type, namely, 290029, 290092, 920029, and 920092. The smallest of them is 290029, which is the number we are looking for. We note that it is not necessary to search for numbers with seven or more digits, as they are all larger than 290029.

c) A first idea is to find a multiple of 2009 that ends in 3, which is immediate: \(7 \times 2009 = 14063\). This is not the answer we are looking for, as 14063 does not hide 2009. But 200900000 is a multiple of 2009, and then

\[
200914063 = 200900000 + 14063 = 100000 \times 2009 + 7 \times 2009 = 100007 \times 2009
\]

is a multiple of 2009 that hides 2009 and ends in 3.