ব্যাংক ডেইলি কুইজ [লং কোর্সের অংশ]

Question: The difference between the greatest and least prime numbers which are less than 70 is- 

Solution:
Greatest prime number (less than 70) = 67
Least prime number = 2
So, their difference = 67 - 2 = 65

Question: Three numbers are in a ratio of 3 : 5 : 6 and their L.C.M is 2400. Their H.C.F is - 

Solution: 
ধরি,
সংখ্যাগুলো যথাক্রমে ৩ক, ৫ক এবং ৬ক
∴ তাদের ল.সা.গু = ৩০ক
এবং তাদের গ.সা.গু = ক

প্রশ্নমতে,
৩০ক = ২৪০০
⇒ ক = ২৪০০/৩০
∴ ক = ৮০

তাহলে, গ.সা.গু = ৮০

Question: The sum of first 55 natural numbers is -

Solution: 
We know that,
The sum of the first n natural numbers = n(n+1)​/2

Sum of first 55 natural numbers is = 55(55 + 1)/2
= (55 × 56)/2
= 1540

Question: The H.C.F and L.C.M of two numbers are 5 and 150 respectively. If one of the numbers is 15, the other one is - 

Solution: 
আমরা জানি,
দুইটি সংখ্যার ল.সা.গু ও গ.সা.গু এর গুণফল সংখ্যা ২টির গুণফলের সমান।

ধরি,
অপর সংখ্যাটি = ক

প্রশ্নমতে,
ক × ১৫ = ৫ × ১৫০
বা, ক = (৫ × ১৫০)/১৫
∴ ক = ৫০

Question: Which one of the following is not a prime number?

Solution: 
69 = 3 × 23 which is not a prime number.
29, 79, 89 are prime numbers.

Question: What is the greatest number of four digits, which, when divided by 6, 12, and 18, leaves a remainder of 3 in each case?

Solution: 
the largest four-digit number is = 9999
the L.C.M of 6, 12, 18 is = 36
dividing 9999 by 36 we get the remainder of 27

so, the number is = (9999 - 27) + 3 = 9975

Question: If 8826P is divisible by 9, what is the value of P? 

Solution:
 একটি সংখ্যা ৯ দ্বারা বিভাজ্য হবে যদি সংখ্যাটির অঙ্কগুলোর সমষ্টি ৯ দ্বারা বিভাজ্য হয়। 

৮ + ৮ + ২ + ৬ = ২৪; এর সাথে ৩ যোগ করলে ২৭ হয়, যা ৯ দ্বারা বিভাজ্য।
∴ P = ৩

Question: Find the greatest number that will divide 39, 87, and 179 and leave the same remainder.

Solution: 
the number is the H.C.F of (87 - 39), (179 - 87) and (179 - 39)
= H.C.F of 48, 92 and 140
= 4

Question: The product of two numbers is 168 and the sum of their square is 289. The sum of the numbers is:

Solution:
Let
the numbers be x and y

Then,
xy = 168
and x² + y² = 289

∴ (x + y)² = x² + y² + 2xy
= 289 + (2 × 168)
= 289 + 336
= 625

∴ x + y = √625 = 25

Question: The H.C.F. of two numbers is 11 and their L.C.M. is 5566. If one of the numbers is 253, then the other is:

Solution:
We know that,
L.C.M × H.C.F. = Product of two numbers
⇒ 5566 × 11 = 253 × other number
⇒ Other number = (5566 × 11) ÷ 253
∴ Other number = 242

Question: Find the square of a positive number which when decreased by 17 is equal to 60 times the reciprocal of the number.

Solution: 
Let the number be x

Then, x - 17 = 60/x
⇒ x² - 17x − 60 = 0
⇒ x² - 20x + 3x − 60 = 0
⇒ (x - 20)(x + 3) = 0
∴ x = 20, - 3 

The positive number = 20
Hence, the square of the positive number = 400

Question: If the product of three consecutive integers is 210, then the sum of the integers is:

Solution:
Given,
the product of three consecutive integers = 210
= 2 × 3 × 5 × 7
= 5 × (2 × 3) × 7
= 5 × 6 × 7

Clearly, the three consecutive integers whose product is 210 are 5, 6 and 7.

∴ Required sum = 5 + 6 + 7
= 18

Question: Find the HCF of 5/6, 3/7 and 11/21 = ?

Solution:
For the HCF of fractions, it has to be taken the HCF of numerators and LCM of denominators.

HCF of 5, 3, 11 = 1
LCM of 6, 7, 21 = 42

HCF of numerators/LCM of denominators = 1/42

Hence, the HCF of 5/6, 3/7 and 11/21 = 1/42

Question: 5 is added to a certain number; the sum is multiplied by 7; the product is divided by 6 and 9 is subtracted from the quotient. Thus, if the remainder left is 12, what was the original number?

Solution: 
let the number be x.

ATQ,
[{7(x + 5)}/6] - 9 = 12
⇒ {7(x + 5)}/6 = 12 + 9
⇒ 7(x + 5) = 21 × 6
⇒ x + 5 = 126/7
⇒ x + 5 = 18
⇒ x = 18 - 5
∴ x = 13

Question: What is the smallest number when added by 5 the summation will be divided by 12, 16, 24 and 32?

Solution:
The required number will be 5 less than L.C.M. of 12, 16, 24 and 32.

12 = 2 × 2 × 3
16 = 2 × 2 × 2 × 2
24 = 2 × 2 × 2 × 3
32 = 2 × 2 × 2 × 2 × 2

So the L.C.M of 12, 16, 24 and 32 = 96

Required number= 96 - 5 = 91