Page 156 - Math Genius C4

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\\September 30, 2025 9:50 AM Bharat Arora P-9 Reader _________________________ Date: ___________________74

Patterns in Multiplication 3. 111 ÷ 3 = 37 4. (9 – 1) ÷ 8 = 1

Observe the pattern in the following multiplications. 222 ÷ 6 = 37 (98 – 2) ÷ 8 = 12
1. 1 × 1 = 1 2. 9 × 1 = 9 333 ÷ 9 = 37 (987 – 3) ÷ 8 = 123
(9876 – 4)
1234
11 × 11 = 121 9 × 11 = 99 444 ÷ 12 = 37 (98765 – 5) ÷ 8 = 12345
÷
÷
555
15
8
=
=
37
111 × 111 = 12321 9 × 111 = 999 666 ÷ 18 = 37 (987654 – 6) ÷ 8 = 123456
1111 × 1111 = 1234321 9 × 1111 = 9999 777 ÷ 21 = 37 (9876543 – 7) ÷ 8 = 1234567
... ... ... ... ... ... 888 ÷ 24 = 37 (98765432 – 8) ÷ 8 = 12345678
Patterns Based on the Multiples of 9 999 ÷ 27 = 37 (987654321 – 9) ÷ 8 = 123456789
Multiply any number by 9; the sum of all the digits of the product is always 9. Patterns in numbers arranged in a shape
1. Multiple of 9 Product Sum of all digits of the product Numbers can be arranged to form a shape or to fit
11 × 9 99 9 + 9 = 18, 1 + 8 = 9 inside a shape. These numbers can also have a pattern. 3 45
21 × 9 189 1+ 8 + 9 = 18, 1+ 8 = 9 Here are some numbers that have been arranged in a 9 37

111 × 9 999 9 + 9 + 9 = 27, 2 + 7 = 9 rectangle. 6 8
2. 12345679 × 9 = 111111111 On observing the figure closely, we find that each 18 7 ? 45

12345679 × 18 = 222222222 Math Insight number in the smaller rectangle is the difference of 11 54
12345679 × 27 = 333333333 If the sum of all the digits of a two numbers outside it.
12345679 × 36 = 444444444 number is 9 or divisible by 9, then the Let us observe these parts individually.
number is called a multiple of 9.
12345679 × 45 = 555555555
casting out 9 3 45 18 7 ? 45

Let us consider the number 952632. It is a multiple of 9 as the sum of its digits 9 6 8 37 11 54

= 9 + 5 + 2 + 6 + 3 + 2 = 27 is a multiple of 9 (2 + 7 = 9), then
952632 952632 522 is also a multiple of 9 (as 5 + 2 + 2 = 9). 9 – 3 = 6; 45 – 37 = 8; 18 – 11 = 7, ... Hence, the missing number = 54 – 45 = 9.

Thus, any number or combination of digits in that number that add to 9 can be cast Think Tank Logical Thinking
out from the given number and the sum Find the number to replace the ‘?’ mark in the given number pattern.

of the number’s remaining digits will also Fast Check
be a multiple of 9. State whether the following numbers 26 28 28 30 30 28 32 30 ?
are multiples of 9, by casting out 9.
Patterns in Division 1. 235897 2. 6810453

Look at the following patterns in division. Practice time 8c

1. 6 ÷ 2 = 3 2. 10 ÷ 5 = 2 1. Find the number patterns and complete them by filling in the missing numbers:

60 ÷ 2 = 30 100 ÷ 5 = 20 (a) 1, 4, 7, 10, 13, _____, _____, _____, _____
600 ÷ 2 = 300 1000 ÷ 5 = 200
6000 ÷ 2 = 3000 10000 ÷ 5 = 2000 (b) 3, 12, 21, 30, _____, _____, _____, _____

60000 ÷ 2 = 30000 100000 ÷ 5 = 20000 (c) 7, 11, 15, 19, 23, _____, _____, _____, _____
... ... ... ... ... ... (d) 512, 256, 128, _____, _____, _____, _____

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