ব্যাংক নিয়োগ প্রস্তুতি ⎯ লং কোর্স

Question: If n is even, (6ⁿ - 1) is divisible by - 

Solution: 
if n = 2 then, (6² - 1) = 35
if n = 4 then, (6⁴ - 1) = 1295
if n = 6 then, (6⁶ - 1) = 46655

the HCF of 35, 1295 and 46655 is 35

so, (6ⁿ - 1) is always divisible by 35

Question: If 2x² + 12x + 18 = 0, what is the value of x?

Solution: 
2x² + 12x + 18 = 0
or, 2(x² + 6x + 9) = 0
or, (x)² + 2.x.3 + (3)² = 0
or, (x + 3)² = 0
or, x + 3 = 0
∴ x = - 3

Question: A number divided by 13 leaves a remainder of 1 and if the quotient, thus obtained, is divided by 5, we get a remainder of 3. What will be the remainder if the number is divided by 65?

Solution: 
ধরি, 
ভাগফল = k
ভাগফলকে 5 দিয়ে ভাগ করলে ভাগশেষ হবে 3.
k কে 5 দ্বারা ভাগ করলে ভাগফল হবে = m

∴ k = 5m + 3

এই ভাগফলের সাথে 13 গুণ করে সাথে 1 যোগ করলে সংখ্যাটি পাওয়া যবে।

∴ সংখ্যাটি = 13(5m + 3) + 1
= 65m + 40

এই সংখ্যাটিকে 65 দ্বারা ভাগ করলে ভাগশেষ হবে 40

Question: A group of 1200 persons consisting of captains and soldiers is travelling in a train. If for every 15 soldiers there is one captain, then the number of captains in the group is - 

Solution: 
let the number of captains = x
ATQ,
x + 15x = 1200
16x = 1200
x = 1200/16
x = 75 

Question: When an integer m is divided by 6, the remainder is 4. What is the remainder when 7m is divided by 3?

Solution:
m = 6 × quotient + remainder
or, m = 6n + 4    [Let, quotient = n]
or, 7m = 7 × 6n + 7 × 4
or, 7m = 7 × 6n + 28

since remainder can not be greater than or equal to divisor
= 6 ×(7n) + 27 + 1
= 3(14n + 9) + 1

if this number is divided by 3 the remainder will be 1

∴ Remainder is 1

Question: The sum of the weights of A and B is 80 kg. Half of the weight of A is equal to 5/6 times the weight og B. Find the weight of B.

Solution: 
A + B = 80 and
A/2 = 5B/6
∴ A = 5B/3

A + B = 80
or, 5B/3 + B = 80
or, 8B/3 = 80
∴ B = 30 Kg

Question: On dividing a number by 5, we get 3 as remainder. What will the remainder when the square of this number is divided by 5?

Solution:
Let,
The number be x
And on dividing x by 5, we get p as quotient and 3 as remainder
∴ x = 5p + 3
or, x² = (5p + 3)²
= 25p² + 30p + 9
= 25p² + 30p + 5 + 4
= 5(5p² + 6p + 1) + 4
∴ On dividing the square of this number by 5, we get the remainder as 4.

Question: Puja has 60 Tk more than Titu and 90 Tk more than Pavel. If their total is 330 Tk, what is Puja's share?

Solution: 
let,
Puja has x Tk.
Titu has (x - 60)Tk.
Pavel has (x - 90)Tk.

x + x - 60 + x - 90 = 330
or, 3x - 150 = 330
or, 3x = 330 + 150
or, x = 480/3
∴ x = 160

So, Puja's share is 160 Tk.

অপশনে দৈত উত্তর থাকায় প্রশ্নটি বাতিল করা হলো।
Question: If p is a prime number greater than 3, then (p² - 1) is always divisible by - 

Solution: 
Let, p = 5
⇒ (p² - 1) = (25 - 1)
⇒ (p² - 1) = 24, which is divisible by 24

Let p = 7
⇒ (p² - 1) = (49 - 1)
⇒ (p² - 1) = 48, which is divisible by 24

Let, p = 11
⇒ (p² - 1) = (121 - 1)
⇒ (p² - 1) = 120, which is divisible by 24

Hence, (p² - 1) is always divisible by 24.

Question: The sum of four consecutive even numbers A, B, C and D is 180. What is the sum of the set of next four consecutive even numbers?

Solution: 
let, the consecutive even numbers are a, a + 2, a + 4, a + 6
∴ a + a + 2 + a + 4 + a + 6 = 180
or, 4a = 180 - 12
or, 4a = 168
∴ a = 42

the numbers are 42, 44, 46, 48.

next 4 consecutive numbers are 50, 52, 54, 56
sum = 50 + 52 + 54 + 56
= 212

Question: If a : b = 4 : 3 then what is the value of (3a + 2b)/(3a - 2b).

Solution: 
a : b = 4 : 3
a/b = 4/3
a = 4b/3

given,
(3a + 2b)/(3a - 2b)
= (4b + 2b)/(4b - 2b)
= 6b/2b
= 3

Question: Find the least number which will leaves remainder 5 when divided by 8, 12, 16 and 20.

Solution: 
We have to find the Least number, therefore we find out the LCM of 8, 12, 16 and 20.
8 = 2 × 2 × 2;
12 = 2 × 2 × 3;
16 = 2 × 2 × 2 × 2;
20 = 2 × 2 × 5;
LCM = 2 × 2 × 2 × 2 × 3 × 5 = 240;
This is the least number which is exactly divisible by 8, 12, 16 and 20
Thus,
Required number which leaves remainder 5 is,
240 + 5 = 245

Question: The sum of two numbers is 45. Their difference is 1/9 of their sum. Their LCM is - 

Solution: 
let the numbers are p and q.
then,
p + q = 45..........(i)
p - q = 45/9 = 5...........(ii)

adding (i) and (ii).
2p = 50
p = 25

putting p on (i) we get,
25 + q = 45
q = 20

the LCM of 25, 20 is 100.

Question: Let x be a number. GCF of 2/5, 3/5 and 1/4 of that number is 5. find the number.

Solution: 
the numbers are 2x/5, 3x/5, x/4.
GCF of 2x, 3x, and x is x
and LCM of 5, 5 and 4 is 20
∴ the GCF of 2x/5, 3x/5, x/4 is x/20

ATQ,
x/20 = 5
x = 100

Question: The LCM and HCF of two numbers are 495 and 5. If the sum of the numbers is 100, what is the difference of the numbers?

Solution: 
let, 
one number is x
∴ another number is (100 - x)

∴ x(100 - x) = 5 × 495
or, 100x - x² = 2475
or, x² - 100x + 2475 = 0
or, x² - 55x - 45x + 2475 = 0
or, x(x - 55) - 45(x - 55) = 0
or, (x - 55)(x - 45) = 0
∴ x = 45, 55
∴ the numbers are 45 and 55.
difference = 55 - 45 = 10

Question: The difference between 4/5 of a number and 40% of the number is 30. what is the 2/5 of that number?

Solution: 
let the number be x.

ATQ,
4x/5 - (40% of x) = 30
or, 4x/5 - 2x/5 = 30
or, 2x/5 = 30
∴ x = 75

∴ 2/5 of 75 is = (75 × 2)/5
= 30

Question: The product and LCM of two numbers are 54 and 18. Find their HCF- 

Solution: 
আমরা জানি,
দুইটি সংখ্যার গুণফল = ল.সা.গু × গ.সা.গু।
গ.সা.গু = দুইটি সংখ্যার গুণফল/ল.সা.গু
গ.সা.গু = ৫৪/১৮
গ.সা.গু = ৩

Question: If x + y : y + z : z + x = 6 : 7 : 8 and x + y + z = 14, find the value of z.

Solution: 
Given,
x + y : y + z : z + x  = 6 : 7 : 8
x + y + z = 14...........(i)

Let,
x + y = 6k, y + z = 7k, z + x = 8k

adding them we get,
2( x + y + z) = 21k
or, 2 × 14 = 21k
or, k = (2 × 14)/21 
∴ k = 4/3

as, x + y = 6k
or, x + y = 6(4/3)
∴ x + y = 8 

from (i) we get,
x + y + z = 14
8 + z = 14 
z = 14 - 8 
z = 6

Question: The product of two numbers is 2028 and their HCF is 13. The number of such pair is - 

Solution: 
let, 
the numbers are 13x and 13y
then,
13x × 13y = 2028
or, xy = 2024/169
∴ xy = 12

number pair that have the product of 12 is (1, 12) and (3, 4)

∴ number-pairs are = (13 × 1, 13 × 12) and (13 × 3, 13 × 4)
= (13, 156), (39, 52)

Question: The sum of two numbers is 14 and their difference is 10. Find the product of these two numbers.

Solution: 
let, the numbers be x and y
∴ x + y = 14.......(i)
x - y = 10......(ii)

adding (i) and (ii) we get,
x + y + x - y = 14 + 10
2x = 24
x = 12

from (i) we get,
12 + y  = 14
y = 2

product = xy = 12 × 2 = 24

Question: The ratio of two numbers is 4 : 5 and their HCF is 4. Their LCM is - 

Solution: 
let, 
the numbers are 4x and 5x
their HCF is = x
∴ x = 4

so, the numbers ar 16, 20.

the LCM is = 80

Question: Three numbers are in a ratio of 4 : 5 : 3. What is the ratio of their H.C.F and L.C.M? 

Solution: 
ধরি,
সংখ্যাগুলো যথাক্রমে ৪ক, ৫ক এবং ৩ক
∴ তাদের ল.সা.গু = ৬০ক
এবং গ.সা.গু = ক

∴ গ.সা.গু : ল.সা.গু = ক : ৬০ক 
= ১ : ৬০

Question: The number of number-pairs lying between 40 and 100 with their HCF as 15 is - 

Solution:
গ.সা.গু ১৫ এমন প্রতিটি সংখ্যার উৎপাদক ১৫।
৪০ থেকে ১০০ এর মধ্যে ১৫ এর গুণিতক ৪৫, ৬০, ৭৫, ৯০

সংখ্যা জোড়: (৪৫, ৬০), (৪৫, ৭৫), (৪৫, ৯০), (৬০, ৭৫), (৬০, ৯০), (৭৫, ৯০)
কিন্তু এখানে (৪৫, ৯০) এর গ.সা.গু হলো ৪৫ এবং (৬০, ৯০) এর গ.সা.গু হলো ৩০।

তাই ৪ টি সংখ্যাজোড় আছে যাদের গ.সা.গু ১৫।

Question: To make 5686 exactly divisible by 7 what number should be subtracted from it??

Solution: 
if we divide 5686 by 7 the remainder is 2
to make it divisible by 7 we need to subtract 2 from it.

so the number is = 5686 - 2 = 5684
dividing 5684 by 7 we get the result = 812