Q) A 2-digit number is seven times the sum of its digits and two (2) more than 5 times the product of its digits. Find the number.

PYQ: 32 (a) – CBSE 2025 – Code 30 – Series 5 – Set 1

Ans:

Step 1: Let’s consider the 2-digits of the number are a and b.

∴ the number is: 10 a + b

Step 2: Let’s make equation from the given 1st condition:

Condition given is “The number is seven times the sum of its digits”

∴ 10 a + b = 7 (a + b)

∴ 10 a + b = 7 a + 7 b 

∴ 10 a – 7 a + b – 7 b = 0

∴ 3 a – 6 b = 0

∴ a = 2 b …………….. (i)

Step 3: Let’s make equation from the given 2nd condition:

Condition given is “The number is two more than five times the product of its digits”

∴ 10 a + b = 2 + 5 (a x b)

∴  10 a + b = 5 a b + 2 …………… (ii)

Step 4: Let’s substitute value of a from equation (i) into equation (ii), we get:

∴ 10 (2 b) + b = 5 (2 b) b + 2

∴  20 b + b = 10 b + 2

∴  21 b = 10 b + 2

∴  10 b – 21 b + 2 = 0 …………….. (iii)

Step 3: Since we can not solve the above quadratic equation by middle term splitting, let’s solve for b by quadratic formula:

We know that the standard quadratic equation is given by: A x² + B x + C = 0

By comparing the quadratic equation (iii) with the standard quadratic equation, we get:

A = 10, B = – 21, C = 2

Let’s calculate the discriminant now:

∴ D = B ² – 4 A C = (- 21) ² – 4 (10) (2)

= 441 – 80 = 361

Next, we calculate value of b in the quadratic equation (iii):

The value of root, b is given by:

b = [- B ± √ (B ² – 4 A C)]  / 2 A

∴ b = [- B ± √ (D)]  / 2 A

∴ b = [ – (- 21) ± √ 361] / 2 (10)               (∵ D = 361 calculated above)

∴ b = (21 ± 19) / 20

This gives us two possible values for b:

1. b = (21 + 19) / 20 = 40 / 20 = 2

2. b = (21 – 19) / 20 = 2 / 20 = 0.1 

Since a and b are the digits of a number, hence, we reject b = 0.1, and accept b = 2 only.

Next, we substitute value of b in equation (i) to get value of a

∴  a = 2 b

∴  a = 2 (2) = 4

Step 4: Now we have caluclated values of both the digits, ie. a = 4 and b = 2

∴ The number ab = 10 a + b = 10 (4) + (2) = 42

Therefore, the number is 42.

Check: By 1st condition, 7 (4 + 2) = 7 x 6 = 42. Since this is equal to the number, hence first condition is satisfied.

By 2nd condition, 2 + 5 (4 x 2) = 2 + 5 x 8 = 42. Since this is equal to the number, hence second condition is also satisfied.

Since both the given conditions are satisfied by our answer, hence our answer is correct.

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