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(b) Clearly, C = 6 because (5 + 6) = 11. So, 1 is carried over. So, the puzzle can be

Then, (1 + 8 + 6) = 15. So, B = 6 and 1 is carried over. solved as below:
6 7 6 5
Next, (1 + 5 + 7) = 13. So, A = 7 and 1 is carried over.
+ 2 5 8 6
Finally, (1 + 6 + 2) = 9. So, D = 2 9 3 5 1

∴ A = 7, B = 6, C = 6, and D = 2.
(c) Here, we observe that the sum of three A’s is a number whose unit digit is A.
This is possible only if A = 0 or 5

If we take A = 0, then the sum is 0 + 0 + 0 = 0 So, the puzzle can be
solved as below:
Then, B also equals zero.
5
But this will make A = B, which is not given. 5

Hence, this possibility is rejected. Therefore, A = 5. + 5
Now, 5 + 5 + 5 = 15. Thus, A = 5 and B = 1. 1 5

Example 5: Find the value of A and B:
A B

× A 7 Quick Check
2 1 1 B Find the value of A and B in the
Solution: Step 1: Here, we have B × 7 = B, this is only possible following multiplication:
2 1
when either B = 0 or B = 5. × B A
Step 2: If A = 1 and B = 0, then 10 × 17 = 170  2110, and 8 B A
if A = 1 and B = 5, then 15 × 17 = 255  2115.
Step 3: If A = 2 and B = 0, then 20 × 27 = 540  2110, and
if A = 2 and B = 5, then 25 × 27 = 675  2115. So, the puzzle can be
solved as below:
Step 4: If A = 3 and B = 0, then 30 × 37 = 1110 ≠ 2110, and 4 5

if A = 3 and B = 5, then 35 × 37 = 1295  2115. × 4 7
Step 5: If A = 4 and B = 0, then 40 × 47 = 1880 ≠ 2110, and 2 1 1 5
if A = 4 and B = 5, then 45 × 47 = 2115.
Thus, A = 4 and B = 5.

Practice Time 15A

1. Write the following numbers in generalised form.

(a) 36 (b) 405 (c) 1211 (d) 6500
2. Write the following in the usual form.
(a) 10 × 6 + 5 (b) 100 × 7 + 10 × 0 + 8 (c) 100 × 0 + 10 × 9 + 9
(d) 1000 × a + 100 × d + 10 × b + c
3. Form all possible 3-digit numbers using all the digits 2, 3, and 7 once and find their sum. Check
whether the sum is divisible by 37 or not. Also, check whether the number 237 is divisible by the
sum of digits of 12.
4. The sum of the digits of a 2-digit number is 12. If the digits are reversed, the new number formed
increases by 36. Find the number.

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