Competive Exam

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• 49 ÷ 9 , remainder 4, 65 ÷ 9 , remainder 2


• Multiply remainders: 4 × 2 = 8, Since 8 < 9, the remainder is 8.


• Verification : 49 × 65 = 3185 (Now divide by 9) = (9 × 353) + 8

Ans: Remainder = 8

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3. Q. Find reminder (1753×1749×83×171) ÷ 17 Options : A. 8  B. 0 C. 7 D. 17

Solⁿ Simple Trick: Take remainders when divided by 17 = 1753 is 2, 1749 is 15, 83 is 15, 171 is 1

Now multiply 2×15×15×1 = 450

450 ÷ 17 reminder is 8,  Ans: A. 8

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4. Q. “What is the remainder when 1!+2!+3!+4!+5!+⋯+1000 is divided by 8 ?

Solⁿ Simple Trick:

• Factorials 4! and above are divisible by 8 , remainder 0.


• Only add 1! + 2! + 3! = 1 + 2 + 6 = 9


• 9 ÷ 8 = remainder 1

Ans: 1

Note:  Why 9÷8 gives remainder 1

• Divide 9 by 8: 9 ÷ 8 = 1 quotient, remainder ?


• 8 * 1 = 8


• 9 − 8 = 1 , this is the remainder

So, 9 ÷ 8 = 1 remainder 1

Rule: Dividend = (Divisor × Quotient) + Remainder , Here: 9 ÷ 8 , 9 = 8×1 + 1. This is why the remainders is 1

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5. Q. What is the remainder when 1!+2!+3!+4!+5!+⋯+1000 is divided by 12 ?

Solⁿ

Trick:

• Factorials 4! and above are divisible by 12 , remainder 0.


• Only Add 1! + 2! + 3! = 1 + 2 + 6 = 9


• 9 ÷ 12 , remainder 9​

Rule: Dividend = (Divisor × Quotient) + Remainder , Ans: 9

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6. Q. Find the remainder when 35¹¹³ is divided by 9 ?

Solⁿ 

Trick: Formula: i. 35^Odd ÷ 9 ⇒ 8 , ii. 35^even ÷ 9 ⇒ 1

Ex: i. 35¹⁵ ÷ 9 → 8 , ii. 35¹⁰⁰ ÷ 9 → 1, 35⁹⁹⁹ ÷ 9 → 8

Exam Rule: Odd exponent ⇒ 8, Even exponent ⇒ 1.​

Example: 35¹¹³ ÷ 9

When dividing by 9:

• 35 → remainder 8


• 8¹ = 8, 8² = 1


• Odd power ⇒ 8, Even power ⇒ 1

113 = Odd , Ans = Remainder = 8

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7. Remainder when 68⁸⁸ ÷ 67 ?

Trick: Whenever the base is one more than the divisor: (Divisor + 1)ⁿ ÷ Divisor , Remainder is always 1

Ex: i. 11⁵⁰÷10 → 1, ii. 26⁹⁹÷25 → 1,  iii. 68⁸⁸÷ 67 → 1, Ans: 1​

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. If a/2 = b/3 = c/5 then find the value of a+b+c / c.​

Option: A. 2  B. 10  C. 12  D. 5

Solⁿ

Given: a/2 = b/3 = c/5​

Ignore the common value and replace: a = 2 , b = 3, c = 5

Now, a + b + c / c = 2 + 3 + 5 / 5 = 10 / 5 = 2

Ans: 2

Trick: Denominators → Numerators

a/2 = b/3 = c/5  ⇒  a + b + c / c = 2 + 3 + 5 / 5 = 2

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Q. Chetan travels from City A to City B. If he drives at 3/5 of his normal speed, he reaches 52 minutes late. How much time (in minutes) does he take to travel from City A to City B at his normal speed ? চেতন A চহৰৰ পৰা B চহৰলৈ যাত্ৰা কৰে। যদি তেওঁ নিজৰ স্বাভাৱিক গতিৰ 3/5​ অংশ গতিত গাড়ী চলায়, তেন্তে তেওঁ ৫২ মিনিট পলমকৈ গন্তব্যস্থানত উপনীত হয়। স্বাভাৱিক গতিত গাড়ী চলালে A ৰ পৰা B লৈ যাবলৈ কিমান সময় লাগিব ?

Options: (A) 69 (B) 83 (C) 78 (D) 82

Solution | সমাধান

• Speed Ratio (গতিৰ অনুপাত) = 5 : 3


• Time Ratio (সময়ৰ অনুপাত) = 3 : 5 (Time is inversely proportional to speed / সময় গতিৰ ব্যস্তানুপাতিক।)


• Difference in time (সময়ৰ পাৰ্থক্য) = 5 − 3 = 2 parts = 52 minutes (৫২ মিনিট)


• 1 part (১ অংশ) = 52 ÷ 2 = 26 minutes (২৬ মিনিট)


• Normal Time (স্বাভাৱিক সময়) = 3 parts = 3 × 26 = 78 minutes (৭৮ মিনিট)

Ans / উত্তৰ: (C) 78 minutes (৭৮ মিনিট)

Trick | কৌশল : Reverse the Speed Ratio to get the Time Ratio. (গতিৰ অনুপাত উল্টাই সময়ৰ অনুপাত পাওঁ।)

Speed 5 : 3 ⇒ Time 3 : 5

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Q. If 7⁵ × 7 × √7 / 7^−5/2 = 7^2x+3 find the value of x. (তেন্তে x-ৰ মান নিৰ্ণয় কৰা।)

Options: (A) 0.5 (B) 0 (C) 3 (D) 2

Solution | সমাধান

Use the law of exponents:

• 7=7¹


• √7 = 7^1/2


• Dividing by 7^−5/2 means adding 5/2 to the exponent. 

7⁵ × 7¹ × 7¹/²  ÷ 7^−5/2  =  7 ^{5 + 1 + 1/2 ​+ 5/2}​ = 7⁹

Since

          7⁹ = 7^2x+3,

equate the exponents : সূচক (ঘাত) সমান ধৰা।

2x + 3 = 9

2x = 6

X = 3

Ans: (C) 3

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Q. Two friends, Kiran and Mahesh, have monthly incomes in the ratio 5 : 8. Each saves ₹72,000 every month. If their monthly expenditure ratio is 2 : 4, find Kiran's monthly income : কিৰণ আৰু মহেশৰ মাহিলী আয়ৰ অনুপাত 5 : 8। দুয়োজনে প্ৰতিমাহে ₹72,000কৈ সঞ্চয় কৰে। যদি তেওঁলোকৰ মাহিলী ব্যয়ৰ অনুপাত 2 : 4 হয়, তেন্তে কিৰণৰ মাহিলী আয় কিমান ?

Options: (A) ₹1,79,000 (B) ₹1,80,000 (C) ₹2,52,000 (D) ₹1,81,000

Trick | কৌশল

Since both save the same amount (₹72,000), use this trick:

• Income Ratio (আয়ৰ অনুপাত) = 5 : 8


• Expenditure Ratio (ব্যয়ৰ অনুপাত) = 2 : 4

Multiply the income ratio by 2 (difference of expenditure ratio): 5 × 2 = 10, 8 × 2 = 16

Now subtract the expenditure parts: 10 − 4 = 6, 16 − 8 = 8

Difference: 8 − 6 = 2 parts = ₹72,000

So,

1 part = 72,000 / 2 = ₹36,000

Kiran's income: 5 × 36,000 = ₹1,80,000

Ans / উত্তৰ: (B) ₹1,80,000

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