When a 3 digit number 984 is added to another 3 digit number 4a3 we get a four digit number 13b7 which is divisible by 11 find the value of a B?

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NUMBER SYSTEM

1. Natural Numbers: These are the numbers (1, 2, 3, etc.) that are used for counting.

2. Whole Numbers: The set of numbers that includes all natural numbers and the number zero are called whole numbers. Whole numbers are also called as Non-negative integers.

3. Prime Numbers: A natural number larger than 1 is a prime number if it does not have other divisors except for itself and 1. (The lowest prime number is 2. 2 is also the only even prime number. The lowest odd prime number is 3.)

4. Composite Numbers: It is a natural number that has at least one divisor different from unity and itself.

5. Even Numbers: An even number is an integer that can be divided by two and remain an integer or has no remainder.

6. Odd Numbers: An integer that is not an even number is an odd number.

7. If the numbers n1 and n2 are exactly divisible by the same number x, then x is a common divisor of n1 and n2. The highest of all the common divisors of n1 and n2 is called as the GCD or the HCF.

8. Co-prime numbers are any two numbers which have an HCF of 1. (Two consecutive natural numbers are always co-prime. Two consecutive odd numbers are always co-prime. Two prime numbers are always co-prime.)

9. HCF of two or more fractions is given by: HCF of numeratorsLCM of denominators

10. LCM of two or more fractions is given by: LCM of numeratorsHCF of denominators

11. LCM × HCF = Product of two numbers (n1 × n2)

12. N =paqbrc, where, p, q and r are prime factors of the number n.

(i) a, b and c are non-negative powers/exponents

(ii) Number of factors of N = (a + 1)(b + 1)(c + 1)

(iii) Number of odd factors will be all possible combinations of powers of odd numbers (excluding any power of 2)

1. Divisibility by 2 or 5: A number is divisible by 2 or 5 if the last digit is divisible by 2 or 5.

2. Divisibility by 3 (or 9): All such numbers the sum of whose digits are divisible by 3 (or 9) are divisible by 3 (or 9).

3. Divisibility by 4: A number is divisible by 4 if the last 2 digits are divisible by 4.

4. Divisibility by 6: A number is divisible by 6 if it is simultaneously divisible by 2 and 3.

5. Divisibility by 8: A number is divisible by 8 if the last 3 digits of the number are divisible by 8.

6. Divisibility by 11: A number is divisible by 11 if the difference of the sum of the digits in the odd places and the sum of the digits in the even places is either zero or is divisible by 11.

7. Divisibility by 12: All numbers divisible by 3 and 4 are divisible by 12.

8. Divisibility by 7, 11 or 13: The integer n is divisible by 7, 11 or 13 if and only if the difference of the number of its thousands and the remainder of its division by 1000 is divisible by 7, 11 or 13.

Products:

Odd × odd = odd

Odd × Even = Even

Even × Even = Even

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Discussion :: Numbers - General Questions (Q.No.37)

Numbers - Important Formulas
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37.

A 3-digit number 4a3 is added to another 3-digit number 984 to give a 4-digit number 13b7, which is divisible by 11. Then, (a + b) = ?

[A].	10
[B].	11
[C].	12
[D].	15

Answer: Option A

Explanation:

4 a 3 | 9 8 4 } ==> a + 8 = b ==> b - a = 8 13 b 7 |

Also, 13 b7 is divisible by 11

(7 + 3) - (b + 1) = (9 - b)

(9 - b) = 0

b = 9

(b = 9 and a = 1)

(a + b) = 10.

Md. Mahbubur Rahaman Sheikh said: (Jul 17, 2010)
How can you got (9 - b) I cannot understand. Please, Explain

BhargaV said: (May 30, 2011)
-> (7 + 3) - (b + 1) simplify this u got (9 - b)

Soma said: (Jun 15, 2011)
I could not get why 13b7 divisible 11 was represented as (7+3) - (b+1). ?

Kiran said: (Jul 10, 2011)
Soma--> the number is divisible by 11 if the difference between the sum of its digit at odd places and the sum of its digit at even places is either 0 or number divisible by 11. 4 a 3 \| ==> here a can either be 0 or 1 9 8 4 } 13 b 7 \|==> so b can be either 8 or 9 but (7+3) - (b+1) should be divisible by 11 or 0 so b can't be 8. so b is 9 and a is 1. hence (a+b) is 10.

Neelu said: (Sep 21, 2011)
I could not understand this problem. i.e. How we got (7+3) - (b+1) from 13b7 ?

Leoarul said: (Sep 22, 2011)
13b7 to know the divisibility of 11, we have to subtract sum of odd placed numbers with sum of even placed numbers. in 13b7 odd places are 7and3 even places are b and 1. now got it.

Nadeeshani said: (Oct 27, 2011)
How do we get (9 - b) = 0 Please explain.....

Manojit Kar said: (Feb 1, 2012)
->211* 11=2321 now (2+2)-(3+1)=0. simplify u got (9 - b)=0

Sai said: (Oct 27, 2012)
Guys i'm confused, how com we take a+8, is there simple method?

Gannu said: (Aug 7, 2013)

If we add 4a3 to 984 then we can clearly see that the result is such a way that the digits in 100'ths place have no carry. That means a+8 is always less than 9 in order to get 13b7 by comparing the places in addition the 10'ths digits underwent. a+8=b --- (1). Then 13b7 is divisible by 11 so the difference of sets, sum of even places and odd places in 13b7 is. 1+b-10=0 or multiple of 11. Try 0. Then b=9 which is less or equal to 9 hence by eq (1). We can get answer.

If you try other than 11 then this adds up a carry in first addition of two number.

Uma said: (Jun 25, 2014)
Here they give two numbers 4a3 and 983. And they said when they are added together, we get 13b7. It means. 4 a 3. 9 8 4. ------ 13 b 7. -------- When we observe this. If 3+4=7. 4+9=13 then a = 0 or 1. We get b = 8 or 9. And also they said that 13 b 7 is divisible by 11. If we take b = 9 i.e., 1397 it is divisible by 11. So b = 9, a = 1. Now a+b = 9+1 = 10.

Senthil Kumar said: (Aug 20, 2014)
Here a=1 is how declared please explain I can't understand?

Biplab Kumar Halder said: (Oct 31, 2014)
If b = 9 then, 4 a 3 9 8 4 ------- 13 b 7 (4+9 = 13)(a+8 = 9)(3+4 = 7). Let see, a+8 = 9. > a = 9-8. > a = 1.

Anil Sarode said: (Feb 9, 2015)
Just use the concept of divisibility of 11 and get the answer. As number is 13b7 is divisible by 11 the difference of alternate digit must be zero or 11, so 1+b = 3+7 or (1+b) - (3+7) = 0. So you got b = 9. A+8 = b, a+8 = 9 so a = 1, hence a+b = 10.

Mukesh Kumar said: (Feb 19, 2015)
By divisibility rule of 11, we know that it's the difference between sum of number at odd places and sum of no. at even places must be zero or divisible by 11. Hence we have, 7+3-(b+1), if we put 9 in place of b we get zero hence the condition is satisfied and the no. Is divisible by 11. So the no. is 1397. Now subtract 984 from 1397, we get 1397-984 = 413. Hence a=1 and b=9. And a+b = 10.

Raja said: (Jun 21, 2015)
Any one please explain the logic.

Aishu said: (Jul 21, 2015)
Luckily in this prob, no formula/logic is required. 4+3=7. 8+a=b. 9+4=13. Since 9+4 = 13 which means there was no carry from previous. A carry will happen only if 8+ a >= 10. i.e a = or > 2. So a has to be 1! When a is 1.8+1 = 9! a=1. b=9.

Yash said: (Jul 21, 2015)
Nice explanation dudes.

Samyuktha said: (Aug 11, 2015)
4 a 3 +9 8 4 ---------- 1 3 b 7 ---------- If any any number added to 8 which is greater than 2 or equals to 2 gives a carry so choose the number in place of a which is less than 2. 8+a = 8+1 = 9 which doesn't give a carry so a=1. 4 1 3 +9 8 4 --------- 1 3 9 7 -----> which is divisible by 11. ----------- a=1, b=9. a+b=1+9=10.

M.devi said: (Aug 11, 2015)
I couldn't understand this problem please say it in another method.

Mohit Bansal said: (Aug 30, 2015)

For this you see the question as a primary school student: Remember one rule just: For a number to be divisible by 11 the sum of odd placed minus the sum of even places should either be zero (0) or 11. Example: abcdef - is divisible by 11 or not. So, check (a+c+e) - (b+d+f) = so what ever you will get should either be zero (0) or 11. Now, the number was. 4 (hundreds) a (tens) 3 (unit) since 4a3. 9 (hundreds) 8 (tens) 4 (unit) 984. __________________________________ ______. 13 (hundreds) b (tens) 7 (unit) 13b7. Look- 3+4 in unit place gives no carry. Now forget middle place and see. 4+9 =13 (no carry came from tens place its visible). So 8 in the tens place can add only 2 numbers which can't give carry they are:. 0 as 0+8=8. 1 as 1+8=9. Rest all number will give carry. Sp finally at last you have two numbers for 4a3. Those are 4{0) 3 or 4{1}3 by adding them both with 984 seperately you will get 1387 and 1397. Check 1387 = abcd = (a+c) - (b+d) => (1+8) - (3+7) => 9-10 = -1 (not the answer). 1397 = abcd = (a+c) - (b+d) => (1+9) - (3+7) => 10-10 = 0 (we needed either zero (0) or 11 we got zero (0). Hence a is 1, so b will be 1+8=9.

So answer is a+b = (1+9) =10.

Swati said: (May 4, 2016)
13b7 must be divisible by 11. First, divide 13 by 11 we get remainder 2. Now divide 2b7 by 11. As the value of b is between 0 to 9 quotient must be 2. Now We want such no which must be 77 which is divisible by 11. Hence the value of b = 9. As b = a + 8, We get the value of a is 1. Hence a + b = 1 + 9 = 10.

Ancy said: (Sep 3, 2016)
If the number 5728abc (0<=a, b, c<=9) is exactly divisible by 12, then find the maximum value of a-b+c? Ans: 4. I didn't understand. Will anyone explain me?

Mir Rokon Uddin said: (Jan 10, 2017)
b - a = 8. Both a and b must be a single digit(they must come from1 to 9). From 1 to 9 if we place a= 9 and b= 1 then the outcome will be 8. So a+b= 10.

Vishal Chauhan said: (Jun 16, 2017)
Thanks @Uma.

Ketty said: (Jul 1, 2017)

By placing b=9, yes we can divide the number by 11. In other question, that I have come across, another way to find whether a number is divisible by a given number is by this method: eg: 517*324=5+1+7+x+3+2+4=22+x; To let the number be divisible by 3, we simply put x=2 to give 24 which is divisible by 3.

Now, for this 1397=1+3+9+7=20; 20 is not divisible by 11. Is this method applicable for only certain questions? Because this clearly doesn't work for this one.

Md. Saifuzaman said: (Jul 26, 2017)
Thank you @Uma.

THE REAL GANGSTA ( SIMPLEST METHOD) said: (Aug 5, 2017)
+984 ------- 13b7 The first thing is to figure out the possible values of the sum. The missing digit can be 0 to 9: 1307, 1317, 1327, 1337, 1347, 1357, 1367, 1377, 1387, 1397. Only one of these is evenly divisible by 11, namely 1397, so b = 9. Filling that in, we have: . 4a3 +984 ------- 1397 From that, it is easy to figure the top number is 413 and thus a = 1. a=1 b=9 a+b = 10 Answer: 10

Kanchan said: (Sep 30, 2017)
Here 4 a 3 And 9 8 4 Gives 1 3 b 7 Since 4+9=13 and at last, we are getting 13 it means the previous no has not generated any carry which can only be possible when a=1. as then we will get 8+1=9. Therefor a=1, b=9. a+b=10.

Saini Ji said: (Nov 11, 2017)
But what about condition that 13b7 is divisible by 11.

Prudvj said: (Feb 8, 2018)
4 digit number 13b7 i. e divided 11. So even-odd then :(3+7)-(1+b )apply b=9. 10-10=0. a+b=1+9=10.

Soni said: (Feb 9, 2018)
Guys, 11 divisibility rule is applicable here so solve it by that rule. It will be more easy.

Chitra said: (Sep 20, 2018)
Nice explain @Samyuktha.

Nandini B N said: (Jul 9, 2020)
To know the number is divisible by 11, the rule is, Sum of all even places=sum of all odd places. In 13b7, 3 + 7 = 1 + b. b=9. Then a+8 = 9. a = 1. a+b = 1 + 9 = 10.