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(a) Complete the three steps for each 2-digit number in the table.
STEP 1 12 13 14 15 16 17 18
STEP 2 21 31 41 51
STEP 3 18 45 63
(b) Complete this table of 2-digit numbers and their reverse differences.
Use part (a) and any patterns you notice to help you.
[Table_1]
(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 part (b) is extended to the right.
These two columns are part of the extended table.
[Table_2]
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.
550
557
542
(a) Complete this table.
You may use any patterns you notice to help you.
[Table_1]
Number | Reverse difference | Number | Reverse difference | Number | Reverse difference
100 | 99 | 110 | | 120 | | 130 | 99
101 | 0 | 111 | 0 | 121 | 0 | 131 | 0
102 | | 112 | | 122 | | 132 | 99
103 | 198 | 113 | | 123 | | 133
104 | 297 | 114 | | 124 | | 134
105 | 396 | 115 | | 125 | | 135
106 | | 116 | | 126 | | 136 | 495
107 | | 117 | | 127 | | 137 | 594
108 | 693 | 118 | | 128 | | 138 | 693
109 | 792 | 119 | | 129 | | 139 | 792
(b) Complete the statement with the largest number possible.
The reverse difference is always a multiple of .................................................
(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.
..................................................................
552
594
532
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.
569
596
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]
515
538
574
