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(a) Complete the three steps for each 2-digit number in the table.
[Table_1]
$$\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline & 12 & 13 & 14 & 15 & 16 & 17 & 18 \\ \hline \text{STEP 1} & & & & & & & \\ \hline \text{STEP 2} & 21 & 31 & 41 & 51 & & & \\ \hline \text{STEP 3} & 18 & & & & 45 & 63 & \\ \hline \end{array}$$
(b) Complete this table of 2-digit numbers and their reverse differences. Use extit{part (a)} and any patterns you notice to help you.
[Table_2]
$$\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \text{Number} & \text{Reverse difference} & \text{Number} & \text{Reverse difference} & \text{Number} & \text{Reverse difference} & \text{Number} & \text{Reverse difference} \\ \hline 10 & 9 & 20 & 18 & 30 & 27 & 40 & 36 \\ \hline 11 & 21 & 9 & 31 & 18 & 41 & 27 \\ \hline 12 & 22 & & 32 & 9 & 42 & \\ \hline 13 & 18 & 23 & 9 & 33 & & 43 & \\ \hline 14 & 24 & & 34 & & 44 & \\ \hline 15 & 25 & 27 & 35 & & 45 & \\ \hline 16 & 45 & 26 & 36 & 36 & 46 & \\ \hline 17 & 27 & 45 & 37 & 36 & 47 & \\ \hline 18 & 63 & 28 & 54 & 38 & 45 & 48 & \\ \hline 19 & 29 & 63 & 39 & 54 & 49 & 45 & \\ \hline \end{array}$$
(c) Find how many 2-digit numbers have a reverse difference of 0.
..................................................
(d) (i) Complete the statement with the largest number possible.
The reverse difference is always a multiple of ..................................................
(ii) The table in extit{part (b)} is extended to the right. These two columns are part of the extended table.
[Table_3]
$$\begin{array}{|c|c|} \hline \text{Number} & \text{Reverse difference} \\ \hline \text{A} & \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline & 9 \\ \hline \end{array}$$
What is the value of the 2-digit number \( A \)?
..................................................
(iii) 64 has a reverse difference of 18. Show how you can use 6 and 4 to work out the reverse difference without using the STEPs.
500
560
573
(a) Complete this table.
You may use any patterns you notice to help you.
[Table_1]
(b) Complete the statement with the largest number possible.
The reverse difference is always a multiple of ................................. [1]
(c) A 3-digit number $abc$ has first digit $a$, second digit $b$ and third digit $c$.
In this part $a > c$.
So the number 601 has $a = 6$, $b = 0$ and $c = 1$.
(i) There is a relationship between $a - c$ and the reverse difference.
Investigate this relationship, giving three examples. Write down this relationship.
................................................................................................................................. [3]
(ii) Anna says that $a$ is the hundreds digit, $b$ is the tens digit and $c$ is the units digit.
She says the value of $abc$ is $100a + 10b + c$.
Anna writes the value of the reverse number $cba$. $100c + 10b + a$
She writes the difference between the two numbers. $(100a + 10b + c) - (100c + 10b + a)$
Continue Anna’s working to show that the reverse difference is $99(a - c)$. [2]
(iii) A 3-digit number has $a = 8$ and a reverse difference of 594.
Find three possible 3-digit numbers.
................................................................................................................................. [2]
552
541
511
Anna’s working in Question 2(c)(ii) is for the 3-digit number $abc$. Anna uses similar working for the 2-digit number $ab$ where $a > b$.
Use Anna’s working to show that your answer to Question 1(d)(i) is correct.
544
593
598
A 4-digit number $abcd$, where $a > d$, has a value of $1000a + 100b + 10c + d$.
(a) Find an expression, in terms of $a, b, c$ and $d$, for the reverse difference. Your answer should have four terms. [3]
(b) A 4-digit number has first digit 7 and last digit 5. Its reverse difference is 2178. Find the connection between the middle two digits of the 4-digit number. [4]
555
524
575
