Number System

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Number System

Q. The sum and difference of two numbers are 72 and 28. Find the product of the numbers. (প্ৰশ্নঃ দুটা সংখ্যাৰ যোগফল 72 আৰু পাৰ্থক্য 28। সংখ্যাদ্বয়ৰ গুণফল নিৰ্ণয় কৰা।)

Solⁿ

Formula / সূত্ৰ: Product = (Sum + Difference)(Sum − Difference) / 4​

গুণফল = (যোগফল + পাৰ্থক্য)(যোগফল − পাৰ্থক্য) / 4

Given, Sum = 72, Difference = 28

Product = (72 + 28)(72 − 28) / 4​ = 100 × 44 / 4 = 4400 / 4 $$=1100$$

​Product of the numbers = 1100 (সংখ্যাদ্বয়ৰ গুণফল = 1100)

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Super Fast Trick / অতি দ্ৰুত কৌশল

First find the two numbers: 72 + 28 / 2 = 50 , 72 − 28 / 2 = 22

Now, 50 × 22 = 1100

Ans = 1100

Trick: I. Number 1 = (Sum + Difference) ÷ 2, II. Number 2 = (Sum − Difference) ÷ 2

Then multiply them.

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Q. How many prime numbers are there between 2 and 32, and what are they ?
Solⁿ:
There are 10 prime numbers between 2 and 32.
They are 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31.

A prime number is a number that can be divided only by 1 and itself.
Ans: 11

Q.  Which of the following numbers is exactly divisible by both 9 and 11 ?
(a) 277218 (b) 10098 (c) 12345 (d) 181998

Ans: (b) 10098

Soln,

• Rule for divisibility by 9: If the sum of digits is divisible by 9, the number is divisible by 9.
  = 1 + 0 + 0 + 9 + 8 = 18 = 18 ÷ 9 = 2 → divisible.


• Rule for divisibility by 11: If the difference between the sum of digits at even and odd places is 0 or a multiple of 11, the number is divisible by 11.
  = (1 + 0 + 8) − (0 + 9) = 9 − 9 = 0 → divisible.
  Hence, 10098 is divisible by both 9 and 11.

Q. Which of the following numbers is NOT divisible by 9 ?
(a) 49104 (b) 77832 (c) 35253 (d) 45390

Ans: (d) 45390

Soln,
Rule for divisibility by 9 → The sum of digits must be divisible by 9.

·   (a) 4 + 9 + 1 + 0 + 4 = 18 → divisible

·   (b) 7 + 7 + 8 + 3 + 2 = 27 → divisible

·   (c) 3 + 5 + 2 + 5 + 3 = 18 → divisible

·   (d) 4 + 5 + 3 + 9 + 0 = 21 → not divisible by 9

Therefore, 45390 is not divisible by 9.

Q. Which of the following numbers is NOT divisible by 8 ?
(a) 35792 (b) 35112 (c) 35412 (d) 35552

Ans: (c) 35412

Soln.
Rule for divisibility by 8 → The number formed by the last three digits must be divisible by 8.

• (a) 792 ÷ 8 = 99 → divisible


• (b) 112 ÷ 8 = 14 → divisible


• (c) 412 ÷ 8 = 51.5 → not divisible


• (d) 552 ÷ 8 = 69 → divisible

Hence, 35412 is not divisible by 8.

Q. If a 7-digit number 5045A1B is divisible by 11, then what can be the value of (A + B)? 

(a) 11 (b) 5 (c) 17 (d) 8

Soln.

Rule of 11:

A number is divisible by 11 if
(Sum of digits at odd places – Sum of digits at even places) = 0 or 11 or multiple of 11.

Number: 5 0 4 5 A 1 B

Odd places → 5 + 4 + A + B = 9 + A + B
Even places → 0 + 5 + 1 = 6

Difference = (9 + A + B) – 6 = A + B + 3

For divisibility by 11,
A + B + 3 = 11
⟹ A + B = 8

Ans: (d) 8

Q. If number 88p554085k6 is divisible by 72, then what is the value of (3k + 2p)?

(a) 12 (b) 7 (c) 13 (d) 23

Ans: (c) 13

Solⁿ.

If number 88p554085k6 is divisible by 72, then what is the value of (3k + 2p)? (a) 12 (b) 7 (c) 13 (d) 23 simple easy trick

1^st→ 72 = 8 × 9

So the number must be divisible by 8 and 9 both.

2^nd→ Rule of 8

Look at the last 3 digits → 5k6
A number is divisible by 8 if the last three digits of the number form a number that is divisible by 8.

 536 ÷ 8 = 67 → exact → so k = 3

3rd → Rule of 9

A number is divisible by 9 if the sum of all its digits is divisible by 9.

Sum of all digits = 8 + 8 + p + 5 + 5 + 4 + 0 + 8 + 5 + 3 + 6 = 52 + p

For divisibility by 9 → Next multiple after 52 is 54,

so, 52 + p = 54

  → p = 54 - 52 = 2

4^th → Find (3k + 2p)

= 3(3) + 2(2)
= 9 + 4 = 13

Ans: (c) 13

Short Trick:

72 = 8 × 9 → check both rules. Last 3 digits 5k6 → 536 ÷ 8 exact ⇒ k = 3.
Sum of digits = 52 + p → next multiple of 9 = 54 ⇒ p = 2. Now (3k + 2p) = 3×3 + 2×2 = 13.
Ans: (c) 13

Q. Which is wrong in the number series below ?  7, 16, 34, 69, 142, 286

Wrong Term: 69

Reason: Series follows ×2 + 2

• 7 → 7×2+2 = 16


• 16 → 16×2+2 = 34


• 34 → 34×2+2 = 70 ⟵ should be 70, not 69


• 70 → 70×2+2 = 142


• 142 → 142×2+2 = 286

Corrected series : 7, 16, 34, 70, 142, 286.

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Q. What is the total number of factors of 840 ?

Solⁿ.

Prime factorization: $$840=$$2³×3¹×5¹×7¹

Number of factors: $$(3+1)(1+1)(1+1)(1+1)$$ = 4 × 2 × 2 × 2 = 32​

Ans: (C) 32

Trick: Add 1 to each power and multiply.

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Q. How many integers are there between 891² and 892² ?

Solⁿ.

For consecutive squares: n² and (n+1)²

Numbers between them: 2n​

Here $n=891$:

        Numbers between them: 2n = 2 × 891= 1782​

Ans: 1782

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Q. How many integers are there between 625² and 626² ?

Solⁿ.

For consecutive squares: n² and (n+1)²

Number of integers between them: 2n​

Here, $n=625$:

Number of integers between them: 2n = 2 × 625 = 1250​

Ans: B) 1250

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Q. How many integers are there between 320² and 321² ?

Solⁿ.

For consecutive squares: n² and (n+1)²

Number of integers between them: 2n​​

Here, $n=320$:

Number of integers between them: 2n = 2 × 320 = 640​

Ans: A) 640

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