Write down any whole number that has at least two digits. Workingfrom left to right, write down the absolute (non-negative) difference ofeach pair of adjacent digits to form a new number. Repeat this processuntil a single digit is reached. That digit is called the digit differenceresidue (DDR) of the original starting number.For example, 153 has a DDR of 2 and 5537 has a DDR of 0 as shown:1 5 34 225 5 3 70 2 42 20Notice, as in the second example, we keep any differences that are 0.This way, the number of digits always decreases by 1 at each step. Alsonote that the original number we start with cannot begin with 0.a Find the nine 3-digit numbers that all have a DDR of 8.b How many 3-digit numbers have a DDR of 0?c Show that two consecutive 3-digit numbers cannot both have a DDRof 0.d Find the smallest pair of consecutive numbers that both have a DDRof 0.
Question
Write down any whole number that has at least two digits. Workingfrom left to right, write down the absolute (non-negative) difference ofeach pair of adjacent digits to form a new number. Repeat this processuntil a single digit is reached. That digit is called the digit differenceresidue (DDR) of the original starting number.For example, 153 has a DDR of 2 and 5537 has a DDR of 0 as shown:1 5 34 225 5 3 70 2 42 20Notice, as in the second example, we keep any differences that are 0.This way, the number of digits always decreases by 1 at each step. Alsonote that the original number we start with cannot begin with 0.a Find the nine 3-digit numbers that all have a DDR of 8.b How many 3-digit numbers have a DDR of 0?c Show that two consecutive 3-digit numbers cannot both have a DDRof 0.d Find the smallest pair of consecutive numbers that both have a DDRof 0.
Solution
a) The nine 3-digit numbers that all have a DDR of 8 are: 189, 279, 369, 459, 549, 639, 729, 819, 909.
b) The 3-digit numbers that have a DDR of 0 are those where all digits are the same. So, there are 9 such numbers: 111, 222, 333, 444, 555, 666, 777, 888, 999.
c) Two consecutive 3-digit numbers cannot both have a DDR of 0. This is because for a number to have a DDR of 0, all its digits must be the same. If we increase or decrease this number by 1 (to get a consecutive number), then at least one of the digits will change, and the DDR will no longer be 0.
d) The smallest pair of consecutive numbers that both have a DDR of 0 are 10 and 11. For 10, the DDR is 0 because the absolute difference between 1 and 0 is 0. For 11, the DDR is also 0 because the absolute difference between 1 and 1 is 0.
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