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A function D(a, 10b + c) is defined as the remainder when the sum a0 + a1….a10b + c is divided by c, where a, b and c are single-digit positive integers.
What is the value of D(y, 10x + z) where x, y and z are single-digit positive integers such that x < y < z ,  x and z are perfect squares and the difference between the sum and the product of the prime factors of y is 1?
  • a)
    0
  • b)
    3
  • c)
    6
  • d)
    7
  • e)
    Cannot be determined
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
A function D(a, 10b + c) is defined as the remainder when the sum a0 +...
Given
  • D(a, 10b + c) =remainder when (a0 + a1….a10b + c ) is divided by c
    • a, b and c are single-digit integers > 0
  • x, y , z are single digit integers > 0
    • x < y < z
    • x = p2 and z = q2, where p, q are integers
    • If y has only one prime factor then, then the difference between the product and the sum of prime factors would be 0
    • So, we can say that y has more than 1 prime factor
To Find: Value of D(y, 10x + z)
  • That is, the value of the remainder when y0+y1…..y10x+z  is divided by z
  • Approach
  • For finding the remainder, we need to find the value of x, y and z
  • Finding value of y
    1. We have inferred that y has more than one 1 prime factor.
    2. Also, we will consider the possible combinations of 2 or more prime factors keeping in the mind the constraint that y lies between 2 distinct single digit perfect squares
      1. There are 3 single digit perfect squares = {1, 4, 9}
  • Finding values of x and z
    1. We are given that x =p2 and z =q2. Also, as x and z are integers < 10, the perfect squares less than 10 can be 1, 4 or 9.
    2. We will apply the constraints given on x and z to determine their values.
  •  
    Working Out
  • Finding value of y
    1. We have inferred that y has more than one prime factor.
    2. Now, the smallest prime factors are {2, 3, 5, 7…}
    3. We will see which combinations of the above prime factors results in a value of y < 10
    4. From the above set, we can see that there is only one possible combination for which the value of y < 10, i.e. y = 2* 3 = 6
    5. So, the product of prime factors of y = 6 and the sum of prime factors of y = 5.
      1. Therefore the difference between the product and the sum of prime factors of y = 1
    6. Hence y = 6
       
  • Finding values of x and z
    1. Perfect squares less than 10 = {1, 4, 9}.  We are not assuming 0 here, as we are given that x, y and z are positive.
    2. As x < z, the possible values of (x, z) = (1, 4) , (1, 9) or (4, 9)
    3. Since y = 6 and x < y < z, (x, z) ≠ (1, 4)
    4. So, (x, z) = (1, 9) or (4, 9)
       
3. Let’s find the value of  D(y, 10x + z) for both the cases of values of (x, z)
a. Case-I: y = 6 and (x, z) = (1, 9)​​​
  • 60+61……619  is to be divided by 9 = 32
  • Since 6=2∗3,62=22∗32
  • So, all the powers of 6 greater than 1, will be divisible by 9
  • Therefore D(6, 10 +9) = Remainder when 60+61(=1+6=7)
  •   is divided by 9 = 7
  • So, D(6,10 + 9) =
​ b. Case-II: y = 6 and (x, z) = (4, 9)
  • 60+61……649  is to be divided by 9 = 32
  • Since 6=2∗3,62=22∗32
  • So, all the powers of 6 greater than 1, will be divisible by 9
  • Therefore D(6, 40 +9) = Remainder when 60+61
  • (= 1 + 6 + 7) is divided by 9 = 7
  • So, D(6, 10 + 9) = 7
  • 4. We see that in both the cases, the value of D(x, 10y+z) = 7.
    Answer: D
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Most Upvoted Answer
A function D(a, 10b + c) is defined as the remainder when the sum a0 +...
To find the value of D(y, 10x z), we need to understand the given conditions and apply them step by step.

Given conditions:
1. x, y, and z are single-digit positive integers.
2. x and z are perfect squares.
3. The difference between the sum and the product of the prime factors of y is 1.

Step 1: Prime factorization of y
We need to find the prime factors of y. Let's assume the prime factors of y are p1 and p2.
So, y = p1 * p2

Step 2: Difference between the sum and product of prime factors
According to the given condition, the difference between the sum and product of the prime factors of y is 1.
So, p1 + p2 - (p1 * p2) = 1

Step 3: Simplifying the equation
We can rearrange the equation as follows:
p1 + p2 = 1 + (p1 * p2)

Step 4: Possible values of p1 and p2
Since p1 and p2 are single-digit positive integers, we can try all possible combinations to satisfy the equation. The possible values are:
p1 = 1, p2 = 1 (1 + 1 = 1 + 1)
p1 = 2, p2 = 1 (2 + 1 = 1 + 2)
p1 = 1, p2 = 2 (1 + 2 = 1 + 2)
p1 = 3, p2 = 1 (3 + 1 = 1 + 3)
p1 = 1, p2 = 3 (1 + 3 = 1 + 3)
p1 = 2, p2 = 2 (2 + 2 = 1 + 4)

Step 5: Checking for the value of D(y, 10x z)
Now, we need to find the remainder when the sum a0 a1.a10b c is divided by c, where a, b, and c are single-digit positive integers.
Given that x, y, and z are single-digit positive integers, we can substitute the values of x and z as perfect squares.

Let's assume the value of x = 1 and z = 4.
So, the expression becomes D(y, 101 4).

Step 6: Calculating the sum and remainder
To find the sum, we add the values of y, 101, and 4.
Sum = y + 101 + 4 = y + 105

Now, we need to find the remainder when the sum is divided by c.
Since the value of c is not given, we cannot determine the exact remainder. Therefore, the answer is e) Cannot be determined.

Note: The answer may be different for different values of x and z, but without knowing the value of c, we cannot determine the exact remainder.
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A function D(a, 10b + c) is defined as the remainder when the sum a0 + a1….a10b + c is divided by c, where a, b and c are single-digit positive integers.What is the value of D(y, 10x + z) where x, y and z are single-digit positive integers such that x < y < z , x and z are perfect squares and the difference between the sum and the product of the prime factors of y is 1?a)0b)3c)6d)7e)Cannot be determinedCorrect answer is option 'D'. Can you explain this answer?
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