Divisibility and Multiples | A

Question 2

Take any 4 consecutive numbers. For example, 3, 4, 5, and 6. Place '++' and '−-' signs in between the numbers. How many different possibilities exist? Write all of them.

Question diagram 1
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Solution
Understand the Question
  • Between any 44 consecutive numbers, there are 33 spaces where a sign can be placed.
  • For each space, there are 22 choices: either a plus sign (++) or a minus sign (−-).
  • The total number of combinations is found by multiplying the choices for each position: 2×2×2=23=82 \times 2 \times 2 = 2^3 = 8.

Step 1 · Calculate the Number of Possibilities

For four numbers (such as 3,4,5,63, 4, 5, 6), there are 33 spaces between them:Question diagram

3 _ 4 _ 5 _ 63 \text{ \_ } 4 \text{ \_ } 5 \text{ \_ } 6

Each space has 22 possible choices (++ or −-):

Total possibilities=2×2×2=23=8\begin{aligned} \text{Total possibilities} &= 2 \times 2 \times 2 \\[0.6em] &= 2^3 \\[0.6em] &= 8 \end{aligned}

Step 2 · List All Possibilities

Systematically listing all 88 sign combinations:

  1. 3+4+5+63 + 4 + 5 + 6
  2. 3+4+5−63 + 4 + 5 - 6
  3. 3+4−5+63 + 4 - 5 + 6
  4. 3+4−5−63 + 4 - 5 - 6
  5. 3−4+5+63 - 4 + 5 + 6
  6. 3−4+5−63 - 4 + 5 - 6
  7. 3−4−5+63 - 4 - 5 + 6
  8. 3−4−5−63 - 4 - 5 - 6
Answer

There are 88 different possibilities:

  • 3+4+5+63 + 4 + 5 + 6
  • 3+4+5−63 + 4 + 5 - 6
  • 3+4−5+63 + 4 - 5 + 6
  • 3+4−5−63 + 4 - 5 - 6
  • 3−4+5+63 - 4 + 5 + 6
  • 3−4+5−63 - 4 + 5 - 6
  • 3−4−5+63 - 4 - 5 + 6
  • 3−4−5−63 - 4 - 5 - 6
Common Mistakes
  • Placing signs in front: Mistakenly placing a sign before the first number 33, which would give 44 spaces (24=162^4 = 16 possibilities) instead of 33 spaces between the numbers.
  • Unsystematic listing: Listing combinations randomly, which often leads to duplicate entries or missed cases.

More questions in A

Q1

Anshu is exploring sums of consecutive numbers. He has written the following:

Now, he is wondering—

  • "Can I write every natural number as a sum of consecutive numbers?"
  • "Which numbers can I write as the sum of consecutive numbers in more than one way?"
  • "Ohh, I know all odd numbers can be written as a sum of two consecutive numbers. Can we write all even numbers as a sum of consecutive numbers?"
  • "Can I write 0 as a sum of consecutive numbers? Maybe I should use negative numbers."

Explore these questions and any others that may occur to you. Discuss them with the class.

Q2

Take any 4 consecutive numbers. For example, 3, 4, 5, and 6. Place '++' and '−-' signs in between the numbers. How many different possibilities exist? Write all of them.

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