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?

Question

Homi · Accepted Answer

Understand the Question

A number is divisible by $8$ if dividing it by $8$ leaves a remainder of $0$ .
Once a multiple of $8$ is found, the next multiple is obtained by adding $8$ .
Any multiple of $1000$ is divisible by $8$ ( $1000 = 8 \times 125$ ). Therefore, whether a large number is divisible by $8$ depends entirely on the number formed by its last three digits.

(i) Find numbers between 120 and 140 that are divisible by 8.

Step 1 · Find Multiples of 8 in the Range

Check the starting number $120$ $120 \div 8 = 15$ Since the remainder is $0$ , $120$ is divisible by $8$ .

Add $8$ successively to find subsequent multiples $120 + 8 = 128$ $128 + 8 = 136$ $136 + 8 = 144$

Since $144 > 140$ , the required numbers are $120, 128,$ and $136$ .

Answer

(i) $120, 128, 136$

(ii) Find numbers between 1120 and 1140 that are divisible by 8.

Step 1 · Find Multiples of 8 in the Range

Check the starting number $1120$ $1120 \div 8 = 140$ Since the remainder is $0$ , $1120$ is divisible by $8$ .

Add $8$ successively to find subsequent multiples $1120 + 8 = 1128$ $1128 + 8 = 1136$ $1136 + 8 = 1144$

Since $1144 > 1140$ , the required numbers are $1120, 1128,$ and $1136$ .

Answer

(ii) $1120, 1128, 1136$

(iii) Find numbers between 3120 and 3140 that are divisible by 8.

Step 1 · Find Multiples of 8 in the Range

Check the starting number $3120$ $3120 \div 8 = 390$ Since the remainder is $0$ , $3120$ is divisible by $8$ .

Add $8$ successively to find subsequent multiples $3120 + 8 = 3128$ $3128 + 8 = 3136$ $3136 + 8 = 3144$

Since $3144 > 3140$ , the required numbers are $3120, 3128,$ and $3136$ .

Answer

(iii) $3120, 3128, 3136$

(iv) What do you observe?

Step 1 · Compare the Three Sets of Numbers

Comparing the numbers found in each range:

Between $120$ and $140$ : $120, 128, 136$
Between $1120$ and $1140$ : $1120, 1128, 1136$
Between $3120$ and $3140$ : $3120, 3128, 3136$

Notice that:

Consecutive numbers divisible by $8$ always have a common difference of $8$ .
The last three digits in each group are identical ( $120, 128, 136$ ). Because $1000$ and $3000$ are both divisible by $8$ , adding thousands does not change the divisibility pattern of the last three digits.

Answer

(iv) Numbers divisible by $8$ are always $8$ apart. Since multiples of $1000$ are divisible by $8$ , the pattern formed by the last three digits remains identical across all three ranges.

Common Mistakes

Excluding the Start Boundary: Assuming that "between $120$ and $140$ " automatically means skipping the endpoints without first checking if $120$ itself is divisible by $8$ .
Confusing Divisibility Rules: Using the divisibility rule of $4$ (checking the last $2$ digits) instead of $8$ (checking the last $3$ digits).

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