Competive Exam Math 7 : Finding the remainder

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8. Finding the remainder when: = 1¹⁵ + 4= 1 + 4 = 5,

Trick

(18¹⁵ + 4) ÷17

• 18 = 17 + 1 (So, replace 18 with 1)


• So, 18 → 1 (when dividing by 17)

Now, 1¹⁵ + 4 = 1+ 4 = 5, Remainder = 5

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Q. Find the remainder when 17²⁴¹ is divided by 18. (Q. 17²⁴¹ ক 18 ৰে ভাগ কৰিলে ভাগশেষ উলিয়াওঁ।)

Solⁿ

We need to find the remainder when : 17²⁴¹ ÷ 18

Since: 17 - 18 = − 1

So, when 17 is divided by 18, the remainder is -1 (or equivalently 17).

Since 241 is an odd number,

So, when divided by 18,

                                 17 ≡ −1 (mod18) 

Now,

• 241 is an odd number.


• Odd power of −1 is −1.

(−1)²⁴¹ = − 1

A remainder cannot be negative, so

−1 = 18 − 1 = 17  [Divisor (ভাজক) - remainder]

Ans: 17

Note:​

• 17 = 18 − 1


• Odd power (বিজোড় ঘাত) → Remainder (ভাগশেষ) = 17


• Even power (যুগ্ম ঘাত) → Remainder (ভাগশেষ) = 1

Ans | উত্তৰ: 17

Trick | ট্ৰিক:

• (n−1)^Odd  ⇒ Remainder = n − 1 


• (n−1)^Even ⇒ Remainder = 1

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Q. What is the remainder when 111²⁰²⁴ is divided by 112 ? Q. 111²⁰²⁴ ক 112 ৰে ভাগ কৰিলে ভাগশেষ কিমান হ'ব ?

Option : A. 3,  B. 2,  C. 4,  D. 1

$$Trick$$

111^{2024 $$÷$$} 112