If 8 - digit number 4432A43B is divisible by 9 and 5, then the sum of A and B is equal to:
12
5
7
8
Concept used
divisibility rule of 9 = sum of all digit should be divisible by 9, then whole number will be divide by 9
divisibility rule of 5 = unit digit should be 0 and 5 , whole number will be divide by 5
Calculation
⇒ according to rule of divisibility of 5 , then B = 0 or 5
⇒ sum of digit when B = 0, sum = 4 + 4 + 3 + 2 + A + 4 + 3 + 0 = 20 + A
⇒ sum of digit , when B = 5 , sum = 4 + 4 + 3 + 2 + A + 4 + 3 + 5 = 25 + A
⇒ we need sum of digit is 27 to divide by 9
⇒ so, sum A and B = 7
If the number 4A306768B2 is divisible by both 8 and 11, then the smallest possible values of A and B will be:
A = 5, B = 3
A = 3, B = 5
A = 5, B = 2
A = 5, B = 4
Concept used:
Divisibility rule of 8 - If the last three digits of a number are divisible by 8, then the number is completely divisible by 8.
Divisibility rule of 11 - If the difference between the sum of the digits at the odd and even places equals 0 or divisible by 11, then the number is divisible by 11.
Calculation:
According to the divisibility rule of 8, we have 8B2 should be divisible by 8
By dividing the number 8B2, the smallest values of B, we put is 3
So, the number is 832, which is completely divisible by 8.
Now,
The updated number = 4A30676832
The difference between the sum of the digits at the odd and even places = (4 + 3 + 6 + 6 + 3) – (A + 0 + 7 + 8 + 2)
⇒ [22 – (A + 17)] = 0 or 11
⇒ A – 5 = 0 or 11
If we put the value of A be 0
Then,
⇒ (A – 5) = 0
⇒ A = 5
∴ The smallest possible values of A and B will be A = 5, B = 3
If the number 1005x4 is completely divisible by 8, then the smallest integer in place of x will be:
SSC CGL 3 March 2020 (Morning)
(a) 2
(b) 4
(c) 1
(d) 0
Ans. (d)
For any number to be divisible by 8, its last 3 digits must be divisible by 8
By putting x=0 in 1005x4, we see that 504 will be divisible by 8.
What should replace * in the number 94*2357, so that the number is divisible by 11?
SSC CGL 3 March 2020 (Evening)
(a) 3
(b) 7
(c) 1
(d) 8
Ans. (a)
For a number to be divisible by 11, the difference of the sum of alternative numbers is divisible by 11.
Hence, (7+3+*+9)-(5+2+4)
=(19+*)-(11)
=8+*
* must be 3 for 94*2357 to be divisible by 11.
When 200 is divided by a positive integer x, the remainder is 8. How many values of x are there?
SSC CGL 3 March 2020 (Afternoon)
(a) 7
(b) 5
(c) 8
(d) 6
Ans. (c)
When 200 is divided by x remainder is 8. So, the number exactly divisible by x is 192.
Multiples of 192 = 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 192
The remainder is always less than the divisor, so :x > 8
Desired values are 12, 16, 24, 32, 48, 64, 96, 192.
When 732 is divided by a positive integer x, the remainder is 12. How many values of x are there?
SSC CGL 4 March 2020 (Morning)
(a) 19
(b) 20
(c) 18
(d) 16
Ans .(b)
When 732 is divided by x remainder is 12. So, the number exactly divisible by x is 720.
The remainder is always less than the divisor, so :x > 12
Now, the factors of 720 which are more than 12 are possible values of x, i.e. (15,16,18,20,24,30,36, 40,45, 48,60, 72, 80, 90, 120, 144, 180, 240, 360, 720).
If the 6-digit numbers x35624 and 1257y4 are divisible by 11 and 72, respectively, then what is the value of (5x-2y)?
SSC CGL 4 March 2020 (Afternoon)
(a) 14
(b) 12
(c) 10
(d) 13
Ans. (a)
In such questions directly check divisibility by 11, 9, and 8.
For a number to be divisible by 11, the difference of the sum of digits at odd or even places must be divisible by 11.
For a number to be divisible by 9, the sum of numbers must be divisible by 9.
For divisibility by 8, the last 3 numbers must be divisible by 8.
Accordingly, For x35624 divisible by 11
(x+5+2)-(3+6+4) = 0 or 11
x=6
And For 1257y4 divisible by 72,
1+2+5+7+y+4 must be divisible by 9 and the only possible value of y is 9, here.
also, 784 is divisible by 8 so the desired value of y = 8
Then, 5x-2y = 30-16 = 14
How many numbers are there from 200 to 800 which are neither divisible by 5 nor by 7?
SSC CGL 4 March 2020 (Evening)
(a) 407
(b) 410
(c) 413
(d) 411
Ans. (d)
Total number from 200 and 800 = 800 – 200 + 1 = 601 ----(including 200)
Total number from 1 to 200 (200 excluding) which are divisible by 5 = 199/5 ≈ 39
Total number from 1 to 800 (800 including) which are divisible by 5 = 800/5 = 160
Total number from 200 to 800 which are divisible by 5 = 160 – 39 = 121
Total number from 1 to 200 (200 excluding) which are divisible by 7 = 199/7 ≈ 28
Total number from 1 to 800 (800 including) which are divisible by 7 = 800/7 ≈ 114
Total number from 200 to 800 which are divisible by 7 = 114 – 28 = 86
Similarly, Total number from 200 to 800 which are divisible by 35 = 800/35 – 200/35 = 22 – 5 = 17
Number neither divisible by 5 nor 7 = 601 – 121 – 86 + 17 = 411
∴ Number neither divisible by 5 nor 7 is 411.
If the nine-digit number 708x6y8z9 is divisible by 99, then what is the value of x+y+z?
SSC CGL 5 March 2020 (Morning)
(a) 9
(b) 16
(c) 5
(d) 27
Ans. (b)
It is given that 708x6y8z9 is divisible by 99.
Thus, 708x6y8z9 is divisible by both 11 and 9
For divisibility by 9, the sum of digits is divisible by 9 (7+0+8+x+6+y+8+z+9 = 38+x+y+z. We get 2 as the remainder when 389. Thus, 2+x+y+z must be divisible by 9)
Possible values of (z+y+x) = 7,16,25, etc.
For divisibility by 11, the difference of sum of digits at odd and even place is divisible by 11 (i.e. in 708x6y8z9 : (9 + 8 + 6 + 8 + 7 ) - (z + y + x + 0) = 38 - ( z + y + x) is divisible by 11 )
Possible values of (z+y+x) =38,5,16 etc.
In such questions, we must directly verify options.
When a positive integer is divided by d, the remainder is 15. When ten times of the same number is divided by d, the remainder is 6. The least possible value of d is:
SSC CGL 5 March 2020 (Afternoon)
(a) 9
(b) 12
(c) 16
(d) 18
Ans. (c)
Let N be the number that gives Q as quotient and 15 as the remainder when divided by d. Thus, d > 15
N = dQ+ 15
10N = 10(dQ)+ 144 + 6
clearly, d is a multiple of 144 which are: 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, and so on.
The least possible value of d is 16. (d > 15)
The greatest number which should be replaced ‘*’ in the number 146*48 to make it divisible by 8 is:
SSC CGL 5 March 2020 (Evening)
(a) 9
(b) 2
(c) 8
(d) 0
Ans. (c)
For 146*48 to be divisible by 8, *48 must be divisible by 8.
Check options: * = 2,8 satisfies the condition. But 8>2. option c is the correct answer.
If the number 687x29 is divisible by 9, then the value of 2x is:
SSC CGL 6 March 2020 (Morning)
(a) 8
(b) 3
(c) 2
(d) 4
Ans. (a)
For 687x29 to be divisible by 9, the sum of digits of 687x29 must be divisible by 9. Thus, x = 4 and 2x = 8
If the number 34k56k is divisible by 6, then what will be the largest value of k?
4
8
6
9
Concept used:
Numbers which are divisible by both 2 and 3 are divisible by 6. That is, if the last digit of the given number is even and the sum of its digits is a multiple of 3, then the given number is also a multiple of 6.
Calculation:
Since k is a digit, the maximum possible value of (K + K) = 18
Now, the sum of the digits of the number = 3 + 4 + k + 5 + 6 + k = 18 + 2k
Here, 34k56k would be divisible by 6, if 2k is divisible by 6.
Now, possible values of 2k are multiples of 6 i.e. 6,12,18.
So, in order to be divisible by 2, the number must have an even number in the unit place.
Hence, the maximum value of 2k = 12 & k = 6
∴ 6 will be the largest value of k.
Which of the following is divisible by 88?
SSC CHSL 2021 (Tier-I) Previous Year Paper (24-May-2022) (Shift 1)
2776408
2776400
2767416
2767440
Concept:
Divisibility test 8 = If the last three digits of a number are divisible by 8, then the number is completely divisible by 8.
Divisibility test 11 = If the difference of the sum of alternative digits of a number is divisible by 11, then that number is divisible by 11 completely.
Calculation:
Number to be divisible by 88 means it should be divisible by both 8 and 11.
From options:
1) 2776408
408 is divisible by 8.
but the difference of the sum of alternative digits of a number is not divisible by 11 so this number is not divisible by 88.
2) 2776400
400 is divisible by 8.
And, the difference of the sum of alternative digits of a number is divisible by 11, so this number is divisible by 88.
3) 2767416
416 is divisible by 8.
but the difference of the sum of alternative digits of a number is not divisible by 11 so this number is not divisible by 88.
4) 2767440
440 is divisible by 8.
but the difference of the sum of alternative digits of a number is not divisible by 11 so this number is not divisible by 88.
∴ 2776400 is divisible by 88.
If a nine-digit number 1263487xy is divisible by both 8 and 5, then the greatest possible values of x and y, respectively, are:
SSC CHSL 2021 (Tier-I) Previous Year Paper (24-May-2022) (Shift 1)
2 and 0
6 and 0
2 and 5
6 and 5
If the number 87m6203m is divisible by 6, then find the sum of all possible values of 'm'?
SSC CHSL 2021 (Tier-I) Previous Year Paper (24-May-2022) (Shift 1)
15
20
10
16
Concept used:
If a number is completely divisible by 3 if the sum of its digits is divisible by 3
If a number ends in a 0, 2, 4, 6 or 8, it is divisible by 2
Calculation:
Possible values of m could be 0, 2, 4, 6, 8
With 0,
8 + 7 + 0 + 6 + 2 + 0 + 3 + 0 = 26 not divisible by 3
With 2
8 + 7 + 2 + 6 + 2 + 0 + 3 + 2 = 30 divisible by 3
With 4
8 + 7 + 4 + 6 + 2 + 0 + 3 + 4 = 34 not divisible by 3
With 6
8 + 7 + 6 + 6 + 2 + 0 + 3 + 6 = 38 not divisible by 3
With
8 + 7 + 8 + 6 + 2 + 0 + 3 + 8 = 42 divisible by 3
So possible values of m is 2 and 8
So, sum = 10
∴ Required answer is 10
If the number 583p2310q2 is divisible by 11, then what is the value of p × q. where p > q?
SSC CHSL 2021 (Tier-I) Previous Year Paper (24-May-2022) (Shift 1)
0
4
6
2
Concept used:
For a number to be divisible by 11, the difference of the sum of digits at even & odd places must either be 0 or equal to 11.
Calculations:
For 583p2310q2
5 + 3 + 2 + 1 + q = 8 + p + 3 + 0 + 2
⇒ 11 + q = 13 + p
⇒ q - p = 2 ( not possible, p > q given)
Condition 2
⇒ 13 + p - (11 + q) = 11
⇒ 13 + p - 11 - q = 11
⇒ p - q + 2 = 11
⇒ p - q = 9
This condition is only possible when, p = 9 and q = 0, as we can put only single digit numbers
So
⇒ p × q = 9 × 0
⇒ 0
∴ The correct choice is option 1
5⁷¹ +5⁷² +5⁷³ +5⁷⁴ +5⁷⁵
is divisible by which of the following number?
71
69
89
73
If the number 55p1067q9 is exactly divisible by 99 then pq is equal to?
(A) 35
(B) 28
(C) 36
(D) 42
If 7A425B is divisible by 36 , then what is the value of A-B ?
(A) 0
(B) 5
(C) 1
(D) 2
If a 9-digit number 5y97405x2 is divisible by 72, then the value of
(x - 2y), for the greatest value of x, is:
(A) 1
(B) 8
(C) 4
(D) 9
The greatest 4-digit number divisible by 36 and 45 is subtracted from 10,000. Find the result.
(A) 150
(B) 100
(C) 180
(D) 120
If the six-digit number 5z3x4y is divisible by 7, 11 and 13, then what is the value of (x + y - z) ?
SSC CGL 2020
4
5
6
3
Which among the given numbers are exactly divisible by 7, 11 and 13?
127721
125127
122227
127127
If the 9 -digit number 83p93678q is divisible by 72, then what is the value of √{p²+q²+12} ?
(A) 6
(B) 7
(C) 8
(D) 9
If a 7-digit number 94x29y6 is divisible by 72, what is the value of {2x+3y} where x is not equal to y?
35
23
21
37
If the 5-digit number 676xy is divisible by 3, 7 and 11, then what is the value of (3x - 5y)?
When 676xy is divisible by 3, 7 &11, it will also be divisible by the LCM of 3, 7 &11.
Dividend = Divisor × Quotient + Remainder
Calculation:
LCM (3, 7, 11) = 231
By taking the largest 5-digit number 67699 and divide it by 231.
∵ 67699 = 231 × 293 + 16
⇒ 67699 = 67683 + 16
⇒ 67699 - 16 = 67683 (completely divisible by 231)
∴ 67683 = 676xy (where x = 8, y = 3)
(3x - 5y) = 3 × 8 - 5 × 3
⇒ 24 - 15 = 9
∴ The required result = 9
If the five-digit number 672 xy is divisible by 3, 7 and 11, then what is the value of (6x + 5y) ?
(A) 16
(B) 23
(C) 17
(D) 24
Ans : (C) 17
672 xy
put x = 2, y = 1
It is divisible by 3, 7, 11
(6x + 5y)
6 x 2+5 x 1 = 12+5 = 17
If a nine-digit number 489 x 6378 y is divisible by 72, then the value of √̅ ̅⁽̅ ̅⁸̅ˣ̅ ̅⁺̅ ̅⁶̅ʸ̅ ̅⁾̅ will be:
Ans : (C) 8
489 x 6378 y, is divisible by 72
72 = is also divisible 9 x 8
x = 5, y = 4 satisfy
so √̅ ̅⁽̅ ̅⁸̅ˣ̅ ̅⁺̅ ̅⁶̅ʸ̅ ̅⁾̅
=√{8×5 + 6×4}=√{64}=8
If the nine-digit number 87605x31y is divisible by 72, then the value of 2x-3y is:
(A) -1
(B) 0
(C) 1
(D) 2
Ans : (D) 2
87605 x 31 y
is divisible by 72
Will also be divisible by 9 x 8 = 72
x = 4, y = 2 satisfy
2x – 3y
2 x 4 – 3 x 2 = 8 – 6 = 2
If the number 4A306768B2 is divisible by both 8 and 11, then the smallest possible values of A and B will be:
(A) A = 3, B = 5
(B) A = 5, B = 4
(C) A = 5, B = 2
(D) A = 5, B = 3
Ans : (D) A = 5, B = 3
4 A 3 0 6 7 6 8 B 2
B = 3 is divisible by 8 [divisible by \frac{832}{8}= ]
The whole number is divisible by 11
A = 5
so A = 5, B = 3
If the 11 digit number 4y6884805x6 is divisible by 72, and x ≠ y , then the value of √̅ ̅ˣ̅ʸ̅ is:
(A) √{12}
(B) √5
(C) √6
(D) √8
Ans : (C) √6
4 y 6 8 8 4 8 0 5 x 6
72 = 9×8, is divisible by 9 and 8
if x = 3 \frac{536}{8}= is divisible
The whole number 4 y 6 8 8 4 8 0 5 3 6 is divisible by 9.
y = 2 Keeping
The Sum of the digits will be divisible by 9 √{xy}= √{3×2}=√6
If the nine-digit number 259876p05 is completely divisible by 11, then what is the value of (p² + 5) ?
(A) 48
(B) 45
(C) 54
(D) 50
Ans : (C) 54
2 5 9 8 7 6 p 0 5
is divisible by 11
Sum of digits at even places – Sum of digits at odd places = 0
(23+P) – 19 = 0 or 11
P = 7 satisfy
P²+5
7²+5 = 54
If the nine-digit number ‘8475639AB’ is divisible by 99, then what is the value of A and B ?
(A) A = 4, B = 6
(B) A = 4, B = 8
(C) A = 3, B = 9
(D) A = 5, B = 3
Ans : (B) A = 4, B = 8
8 4 7 5 6 3 9 A B,
is divisible by 99
Will also be divisible by 99 = 11 x 9
so A = 4, B = 8 satisfy
A = 4, B = 8
If a seven-digit number 7x634y2 is divisible by 88, then for the largest value of y, what is the difference of the values of x and y?
8
4
2
6
The divisibility rule of 8 states that for a number to be divisible by 8, its last three-digit should be perfectly divisible by 8.
The divisibility rule of 11 states that for a number to be divisible by 11, the difference between the sum of odd and even digits should be completely divisible by 11.
Calculations:
Using Divisibility rule of 8, we get-
∴ 4y2 is completely divisible by 8 if y is 7(largest value of y). (∴ y = 7)
Similarly using divisibility rule of 11, we get-
⇒ (7 + 6 + 4 + 2) - (x + 3 + 7) = 19 - 10 - x = 9 - x
⇒ The number is perfectly divisible if x = 9 (∴ x = 9)
∴ Value of x - y = 9 - 7 = 2
The value of x - y is 2.
If a number 5x423y is completely divisible by 88 then find the value of 5x - 8y.
16
24
32
40
Divisibility rule of 8 = Last three digit of any number should be divisible by 8 , then number is divisible by 8
Divisibility Rule of 88 = Number should be divisible by both 8 and 11.
Divisibility Rule of 11 = Sum of odd place digit - Sum of even digit place = 0 or multiple of 11
Calculation
If a number 5x423y is completely divisible by 88
So, By using divisibility rule of 8
y = 2 because 232 is divisible by 8
Now, By using divisibility rule of 11
⇒ (x + 2 + 2) - (5+4+3) = 0
⇒ (x + 4) - (12) = 0
⇒ x = 8
The value of 5x - 8y = 40 - 16 = 24
∴ The required answer is 24
If the nine-digit number 23541y49x is divisible by 72, then (3x + 5y) ∶ (5x + 3y) is equal to:
7 ∶ 9
4 ∶ 3
9 ∶ 7
3 ∶ 4
Concept:
Divisibility by 9 = When the sum of the digits of a number is divisible by 9, the number is said to be divisible by 9.
Divisibility by 8 = When the last 3 digits of a number are divisible by 8, the number is said to be divisible by 8.
Given:
23541y49x is divisible by 72
Calculation:
23541y49x is divisible by 72, which is the product of 8 and 9, which are two co-prime. So the number has to be divisible by both 8 and 9.
23541y49x is divisible by 8, if 49x is divisible by 8, which gives x = 6.
The number becomes 23541y496, which will be divisible by 9, if the sum of the digits becomes divisible by 9.
Now, 2 + 3 + 5 + 4 + 1 + y + 4 + 9 + 6 = 34 + y
⇒ y should be 2, so that the digit sum becomes (34 + 2 = 36) which is divisible by 9.
For x = 6, y = 2
(3x + 5y) ∶ (5x + 3y)
⇒ (3 × 6 + 5 × 2) : (5 × 6 + 3 × 2)
⇒ (18 + 10) : (30 + 6)
⇒ 28 : 36
⇒ 7 : 9
∴ The required result = 7 : 9
The number 3¹³ - 3¹⁰ is divisible by:
2 and 3
3 and 13
2,3 and 13
2,3 and 10
Given:
3¹³ - 3¹⁰
3¹⁰{3³-1}
3¹⁰×26
3¹⁰ × 2× 12
Clearly, in factorization,
2,3,13 are the factor.
The ten digit number 2x600000y8 is exactly divisible by 24. If X ≠ 0 and y ≠ 0, then the least value of (x + y) is equal to:
9
2
5
8
For a number to be divisible by 24,
it should be divisible by 8 and 3.
For a number to be divisible by 8,
the last three digits should be divisible by 8.
For,
y = 4, the last three digits,
i.e, 048 is divisible by 8.
For a number to be divisible by 3,
the sum of the digits should be divisible by 3.
⇒ 20 + x should be divisible by 3
For x = 1, 20 + x = 21 is divisible by 3
⇒ x = 1 and y = 4
∴ x + y = 1 + 4 = 5.
The question has clearly mentioned X ≠ 0 and y ≠ 0
The 10-digit number 79x00001y6 is exactly divisible by 88. What is the value of (x + y)?
7
5
6
9
For the divisible by 88 it will be also divisible by 11 and 8.
For the divisible by 11, difference of sum of alternate digit of a number should be zero.
⇒ (7 + x + 0 + 0 + y) ∼ (9 + 0 + 0 + 1 + 6) = 0
⇒ x + y + 7 = 16
⇒ x + y = 9
For the divisible by 8, last three digits of a number are divisible by 8.
⇒ y = 3, 7
∵ x + y = 9
When y = 3 ⇒ x = 6
When y = 7 ⇒ x = 2
⇒ x + y = 6 + 3 = 2 + 7 = 9
If a 10-digit number 46789x531y is divisible by 72, then the value of (2x + 5y), for the largest value of x is:
38
28
10
16
For divisible by 72, number should be divisible by 2 and 3.so,
⇒ 4 + 6 + 7 + 8 + 9 + x + 5 + 3 + 1 + y = 43 + x + y
We take the maximum value of x = 9
So, 43 + 9 + y = 52 + y
We take the value of y = 2 for divisible by 2 and 3.
⇒ 2x + 5y = 2 × 9 + 5 × 2 = 28
What is the least value of x such that 517x324 is divisible by 12?
A: 0
B: 1
C: 2
D: 5
If 543247x968y is divisible by 90 .Then find the value of 4x + 5y?
A: 30
B: 18
C: 24
D: 36
If the 7 digit no. 56x34y4 is divisible by 72, then the least value of x+y is?
A: 8
B: 12
C: 5
D: 14
If the 8 digit no. 2074x4y2 is divisible by 88, then the value of 4x+3y is?
A: 36
B: 42
C: 54
D: 45
If X381 is divisible by 11, then the digit at the place of X is
(a) 0
(b) 1
(c) 4
(d) 7
Which of the following numbers will completely divide
4¹² + 4¹³ + 4¹⁴ + 4¹⁵
3
7
11
17