Prime Time | IT

Question 31

Consider these statements:

Only the last two digits matter when deciding if a given number is divisible by 4.
If the number formed by the last two digits is divisible by 4, then the original number is divisible by 4.
If the original number is divisible by 4, then the number formed by the last two digits is divisible by 4.

Do you agree? Why or why not?

Solution

We can write any number as a sum of its hundreds part and its last two digits.

Step 1 — How numbers are built

Let us take any number. We can split it into two parts. One part is made of hundreds. The other part is the last two digits. For example, let us take the number 532. We can write 532 as 500 plus 32. The number 500 is $5 \times 100$ . We know that 100 is divisible by 4. $100 \div 4 = 25$ So, any number of hundreds is also divisible by 4. For example, 500 is divisible by 4. $500 \div 4 = 125$ This means the 'hundreds part' is always divisible by 4.

$\boxed{\text{The hundreds part is always divisible by 4.}}$

Diagram 1

Step 2 — Checking Statement 1

Statement 1 says: "Only the last two digits matter when deciding if a given number is divisible by 4." From Step 1, we know the hundreds part is always divisible by 4. So, it does not affect the divisibility by 4. The divisibility of the whole number depends only on the last two digits. If the last two digits are divisible by 4, the whole number is. If not, the whole number is not. So, statement 1 is true.

$\boxed{\text{Statement 1 is True.}}$

Step 3 — Checking Statement 2

Statement 2 says: "If the number formed by the last two digits is divisible by 4, then the original number is divisible by 4." Let us use our example, 532. We know 532 is $500 + 32$ . The hundreds part, 500, is divisible by 4. Let us say the last two digits, 32, are also divisible by 4. $32 \div 4 = 8$ So, we have a number that is (divisible by 4) + (divisible by 4). When we add two numbers that are both divisible by 4, their sum is also divisible by 4. So, 532 is divisible by 4. $532 \div 4 = 133$ This shows statement 2 is true.

$\boxed{\text{Statement 2 is True.}}$

Step 4 — Checking Statement 3

Statement 3 says: "If the original number is divisible by 4, then the number formed by the last two digits is divisible by 4." Let us take a number that is divisible by 4. For example, 724. We know 724 is divisible by 4. $724 \div 4 = 181$ We can write 724 as 700 plus 24. We know 700 is divisible by 4. If 724 is divisible by 4, and 700 is divisible by 4. Then the difference between them must also be divisible by 4. The difference is the last two digits. $724 - 700 = 24$ So, 24 must be divisible by 4. $24 \div 4 = 6$ This shows statement 3 is true.

$\boxed{\text{Statement 3 is True.}}$

Answer

(i) True. Because if the number formed by the last two digits is divisible by 4, the entire number is divisible by 4. (ii) True. Because this is the rule of divisibility by 4. (iii) True. Because if a number is divisible by 4, its last two digits will always form a number that is divisible by 4.

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Q10

In the treasure finding game, a pair of numbers is safe if Jumpy cannot reach both using any jump size other than 1.

Q. Check if these pairs are safe:

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d. 20 and 55

Q11

Which of the following pairs of numbers are co-prime?

a. 18 and 35

b. 15 and 37

c. 30 and 415

d. 17 and 69

e. 81 and 18

Q12

While playing the 'idli-vada' game with different number pairs, Anshu observed something interesting!

Sometimes the first common multiple was the same as the product of the two numbers.
At other times the first common multiple was less than the product of the two numbers.

Find examples for each of the above. How is it related to the number pair being co-prime?

Q13

Multiply back to see that you get 36 in all four cases.

Q14

Using this diagram, can you explain why $30 = 2 \times 3 \times 5$ , no matter which way you multiply 2, 3, and 5?

Q15

Let us take 8560. Does it have any factors from 2 to 10 (2, 3, 4, 5, ..., 9, 10)?

Q16

It is easy to check if some of these numbers are factors or not without doing long division. Can you find them?

Q17

Is 125 a multiple of 10? Will this number appear in the previous sequence? Why or why not?

Q18

Can you now answer if 8560 is divisible by 10?

Q19

Consider this statement:

Numbers that are divisible by 10 are those that end with '0'. Do you agree?

Q20

Explore by listing down the multiples: 5, 10, 15, 20, 25, ... What do you observe about these numbers? Do you see a pattern in the last digit?

Q21

What is the largest number less than 399 that is divisible by 5? Is 8560 divisible by 5?

Q22

Consider this statement:

Numbers that are divisible by 5 are those that end with either a '0' or a '5'. Do you agree?

Q23

The first few multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ... What do you observe? Do you see a pattern in the last digit?

Q24

Is 682 divisible by 2? Can we answer this without doing the long division?

Q25

Is 8560 divisible by 2? Why or why not?

Q26

Consider this statement:

Numbers that are divisible by 2 are those that end with '0', '2', '4', '6' or '8'. Do you agree?

What are all the multiples of 2 between 399 and 411?

Q27

Look at its multiples: 4, 8, 12, 16, 20, 24, 28, 32, ...

Are you able to observe any patterns that can be used? The multiples of 10, 5 and 2 have a pattern in their last digits which we are able to use to check for divisibility. Similarly, can we check if a number is divisible by 4 by looking at the last digit?

Q28

Can we answer the question by looking at more digits? Make a list of multiples of 4 between 1 and 200 and search for a pattern.

Q29

Find numbers between 330 and 340 that are divisible by 4. Also, find numbers between 1730 and 1740, and 2030 and 2040, that are divisible by 4. What do you observe?

Q30

Is 8536 divisible by 4?

Q31

Consider these statements:

Only the last two digits matter when deciding if a given number is divisible by 4.
If the number formed by the last two digits is divisible by 4, then the original number is divisible by 4.
If the original number is divisible by 4, then the number formed by the last two digits is divisible by 4.

Do you agree? Why or why not?

Q32

Find numbers between 120 and 140 that are divisible by 8. Also find numbers between 1120 and 1140, and 3120 and 3140, that are divisible by 8. What do you observe?

Q33

Change the last two digits of 8560 so that the resulting number is a multiple of 8.

Q34

Consider these statements:

Only the last three digits matter when deciding if a given number is divisible by 8.
If the number formed by the last three digits is divisible by 8, then the original number is divisible by 8.
If the original number is divisible by 8, then the number formed by the last three digits is divisible by 8.

Do you agree? Why or why not?

Q35

There are four numbers in this box. Which number looks special to you? Why do you say so?