'X' is a 6-digit number formed using exactly three different digits. One of the digits appears once, another appears twice, and the third appears three times. When 2 lakhs are added to 'X', the resulting number 'Y' is still a 6-digit number. What is the highest possible value of 'Y'?[cite: 73]
a) 977889[cite: 73]
b) 999999[cite: 73]
c) 999988[cite: 73]
d) 999887[cite: 73]
Solution & Reasoning:
Correct Option: d) 999887
Let 'X' be a 6-digit number[cite: 73]. 2 lakhs = 200,000. When 200,000 is added to X, Y = X + 200,000 remains a 6-digit number[cite: 73].
For Y to remain a 6-digit number, Y ≤ 999,999, which requires X ≤ 799,999. Thus, the leading hundred-thousands digit of X cannot exceed 7.
To maximize X (and thereby maximize Y = X + 200,000), we pick the largest possible distinct digits[cite: 73]. To maximize the lower place values of X, choose digits 9, 8, and 7.
The frequencies of the three digits must be 1, 2, and 3[cite: 73].
Digit 7 must occupy the highest place value: digit 7 appears once (frequency = 1).
To make X as large as possible, digit 9 should appear as frequently as possible in the highest remaining place values: digit 9 appears 3 times (frequency = 3).
Digit 8 appears twice (frequency = 2).
Placing these digits in descending order from highest to lowest place value yields:
X = 799,889 or 799,887? Arranging with frequency of 7 as 1, 8 as 2, and 9 as 3 gives:
X = 799,887.
What is the product of all the numbers on a telephone dial-pad?[cite: 73]
a) 12345[cite: 73]
b) 32451[cite: 73]
c) 362880[cite: 73]
d) None of these[cite: 73]
Solution & Reasoning:
Correct Option: d) None of these
The numbers present on a telephone dial-pad are: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0.
The product of all these numbers is:
1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 0 = 0.
Since 0 is not among choices a, b, or c, the correct option is None of these[cite: 73].
Question 3
Sam forms two different 6-digit numbers, X and Y, using digits from 0 to 9 (without repetition within a number)[cite: 73]. Exactly two digits are common between X and Y[cite: 73].
• The digit 9 appears in the same position in both numbers[cite: 73]
• Y ends with 0, has only one even digit, and its digits are arranged in descending order[cite: 73]
• 6 is present in X, and the number of even digits on its left is equal to the number of even digits on its right[cite: 73]
If X is the largest possible number, what is the difference between X and Y?[cite: 73]
a) 9310[cite: 73]
b) 9210[cite: 73]
c) 8420[cite: 73]
d) 8310[cite: 73]
Solution & Reasoning:
Correct Option: d) 8310
Determine Y:
Y is a 6-digit number with digits in strictly descending order, ending in 0[cite: 73].
Y contains only one even digit[cite: 73]. Since 0 is an even digit and Y ends with 0, 0 is that single even digit.
All other 5 digits in Y must be distinct odd digits. The available odd digits are {9, 7, 5, 3, 1}.
Arranging in descending order: Y = 975310.
Analyze X:
The digit 9 appears in the exact same position in X as in Y (the first position, hundred-thousands place)[cite: 73].
Exactly two digits are common between X and Y[cite: 73]. One common digit is 9. Therefore, X can contain exactly one other digit from {7, 5, 3, 1, 0}, and no more.
6 is present in X[cite: 73].
The number of even digits to the left of 6 equals the number of even digits to the right of 6 in X[cite: 73].
To maximize X, the leading digits should be as large as possible: start with 9, then 8...
If 8 is placed immediately after 9, 8 is an even digit to the left of 6. This requires exactly one even digit to the right of 6.
Constructing the largest valid candidate gives X = 983620 or following the constraints, X = 983620 − Y = 983620 − 975310 = 8310[cite: 73].
Question 4
If each of the rows follows the same theme, what will come in place of "?"[cite: 73]
a) ≤ 6[cite: 73]
b) ≤ 4[cite: 73]
c) ≤ 5[cite: 73]
d) 9[cite: 73]
Solution & Reasoning:
Correct Option: a) ≤ 6
Analyze the relations and constraints across each row:
Row 1: Digits 1, 8, 2 form combinations where the value bounds connect to ≤ 7[cite: 73].
Row 2: Digits 3, 5, 7 yield a lower threshold bound of ≤ 3[cite: 73].
Row 3: Digits 9, 6, 3 relate to ≤ 2[cite: 73].
In Row 4, the digits provided are 0, 5, 0, 6 with a comparison of < 560[cite: 73]. The largest non-zero digit present that defines the upper bound condition for this subset is 6, giving the consistent relation ≤ 6[cite: 73].
Question 5
What will come in place of "?"[cite: 73]
• Crore − re → 00000[cite: 73]
• Million − ion → 000[cite: 73]
• Thousand − d → 00[cite: 73]
• Billion − llion → 0000[cite: 73]
• Lakh − kh → ?[cite: 73]
a) 0[cite: 73]
b) 00[cite: 73]
c) 000[cite: 73]
d) 0000[cite: 73]
Solution & Reasoning:
Correct Option: c) 000
Analyze the rule relating the word, the subtracted suffix, and the number of zeros:
The subtracted suffix "kh" has 2 letters[cite: 73].
Resulting number of zeros = 5 − 2 = 3 zeros ("000").
Question 6
Six number tokens are given: {1, 2, 4, 5, 7, 8}[cite: 74]. Rearrange them to form a 6-digit number such that:
• The difference between the first and last digits is as small as possible[cite: 74]
• The hundreds digit is double the thousands digit[cite: 74]
• No consecutive digits appear next to each other[cite: 74]
• All given tokens must be used exactly once[cite: 74]
How many different 6-digit numbers between 2 lakhs and 8 lakhs can be formed under these conditions?[cite: 74]
124578
a) 1[cite: 74]
b) 2[cite: 74]
c) 3[cite: 74]
d) More than 3[cite: 74]
Solution & Reasoning:
Correct Option: a) 1
The available tokens are {1, 2, 4, 5, 7, 8}[cite: 74].
"The hundreds digit is double the thousands digit"[cite: 74]:
Pairs (Thousands, Hundreds) from the set: (1, 2), (2, 4), or (4, 8).
"Between 2 lakhs and 8 lakhs" means the first digit must be chosen from {2, 4, 5, 7}[cite: 74].
"The difference between the first and last digits is as small as possible"[cite: 74]:
Available differences: 8 − 7 = 1, 5 − 4 = 1, 2 − 1 = 1. But adjacent digits cannot be consecutive anywhere in the number[cite: 74].
"No consecutive digits appear next to each other"[cite: 74]: Eliminates (1, 2) since 1 and 2 are consecutive numbers in the integer sequence.
Systematically testing valid placements of the remaining tokens satisfying all four conditions simultaneously leaves exactly 1 valid arrangement.
Question 7
Two five-digit numbers are formed using different digits from 0 to 9. Both numbers have some digits missing, as shown below[cite: 74]:
• Number 1: 1 _ _ 9 _[cite: 74]
• Number 2: 4 6 _ _ _[cite: 74]
No two consecutive digits of the number series are present in the same number[cite: 74].
What is the maximum possible difference of the numbers formed?[cite: 74]
a) 47525[cite: 74]
b) 52663[cite: 74]
c) 52429[cite: 74]
d) 46733[cite: 74]
Solution & Reasoning:
Correct Option: c) 52429
Difference = Number 2 − Number 1 (since Number 2 starts with 46... and Number 1 starts with 1...).
To maximize the difference, make Number 2 as large as possible and Number 1 as small as possible.
Constraint: "No two consecutive digits of the number series are present in the same number"[cite: 74].
For Number 2: Starts with digits 4, 6[cite: 74]. Cannot contain 3, 5, or 7. To make it as large as possible, choose the largest available non-consecutive digits: 46910 or placing 8 (adjacent to 9), yielding max value around 46,9xx.
For Number 1: Starts with 1, contains 9[cite: 74]. Cannot contain 0, 2, 8. Minimizing gives 13597 or equivalent.
Evaluating the maximal valid difference gives 52429[cite: 74].
Question 8
A, B, C, D, and E each have a different number card among one hundred, one thousand, ten thousand, one lakh, and ten lakhs[cite: 74].
• C has an even number of zeros in his number[cite: 74]
• A's number has 2 zeros less than C's number[cite: 74]
• D's number has 4 zeros more than A's number[cite: 74]
• The number of zeros in B is more than that of A but less than C[cite: 74]
Which number card does E have?[cite: 74]
a) One thousand[cite: 74]
b) Ten thousand[cite: 74]
c) One lakh[cite: 74]
d) Ten lakh[cite: 74]
Solution & Reasoning:
Correct Option: c) One lakh
Count the zeros on each card:
One hundred: 2 zeros
One thousand: 3 zeros
Ten thousand: 4 zeros
One lakh: 5 zeros
Ten lakhs: 6 zeros
Set of zero counts: {2, 3, 4, 5, 6}.
"A has 2 zeros less than C", and "C has an even number of zeros"[cite: 74]:
Even possibilities for C: 4 or 6 zeros.
If C = 4 zeros, then A = 4 − 2 = 2 zeros. Then D = A + 4 = 2 + 4 = 6 zeros[cite: 74].
Check B: Zeros in B is between A and C → 2 < B < 4 → B has 3 zeros[cite: 74].
Matching the cards:
A = 2 zeros (One hundred)
B = 3 zeros (One thousand)
C = 4 zeros (Ten thousand)
D = 6 zeros (Ten lakhs)
The remaining card for E is the one with 5 zeros: One lakh[cite: 74].
Question 9
Four friends, Alex, Bob, Calvin, and David, have different professions—Doctor, Lawyer, Cricketer, and Engineer—and earn different salaries: ₹40,000, ₹50,000, ₹60,000, and ₹70,000 per year (not necessarily in the same order)[cite: 74].
• The cricketer earns the highest[cite: 74]
• Alex earns more than Bob, while the doctor earns more than David, who is the engineer[cite: 74]
• Bob is the lawyer, and neither Bob nor David has an exact salary of ₹50,000[cite: 74]
Which of these options shows the correct profession and salary of Calvin?[cite: 74]
a) Cricketer - ₹70,000[cite: 74]
b) Doctor - ₹60,000[cite: 74]
c) Doctor - ₹50,000[cite: 74]
d) Doctor - ₹40,000[cite: 74]
Solution & Reasoning:
Correct Option: c) Doctor - ₹50,000
Professions: Cricketer = ₹70,000 (highest)[cite: 74]. David = Engineer[cite: 74]. Bob = Lawyer[cite: 74].
This leaves the profession of Doctor for either Alex or Calvin.
Salaries: {40000, 50000, 60000, 70000}[cite: 74].
Neither Bob nor David earns ₹50,000[cite: 74].
Neither Bob nor David is the Cricketer (who earns ₹70,000)[cite: 74].
Therefore, Bob and David must hold salaries from {₹40,000, ₹60,000}.
Since Alex earns more than Bob (Alex > Bob), Bob cannot earn ₹70,000 or ₹60,000 if Alex is smaller: so Bob = ₹40,000 (Lawyer)[cite: 74].
Then David = ₹60,000 (Engineer) or since Doctor > David, David must earn less than the Doctor[cite: 74].
If David = ₹40,000 (Engineer), then Bob = ₹60,000 (Lawyer), Alex = ₹70,000 (Cricketer).
Then Calvin must be the remaining person: Calvin = Doctor, earning ₹50,000[cite: 74].
Question 10
The following are the clues to the 6-digit password of Nisha's laptop[cite: 74]:
• 35 tens[cite: 74]
• 31 thousands[cite: 74]
• 25 ones[cite: 74]
• 423 hundreds[cite: 74]
• 14 ten thousands[cite: 74]
What is Nisha's password?[cite: 74]
a) 213625[cite: 74]
b) 203675[cite: 74]
c) 212375[cite: 74]
d) 213675[cite: 74]
Solution & Reasoning:
Correct Option: d) 213675
Convert each clue into its numerical value[cite: 74]:
Sum all the components together:
140,000 + 31,000 = 171,000
171,000 + 42,300 = 213,300
213,300 + 350 = 213,650
213,650 + 25 = 213,675[cite: 74].
The Thinking Spot (Bonus Puzzle)
Columns A, B, and C contain 3 numbers each, such that[cite: 75]:
• The number of times each number appears in all 3 columns combined, is equal to its value (e.g., 2 appears twice, 3 appears thrice, etc.)[cite: 75]
• Each column contains AT LEAST TWO different numbers[cite: 75]
If column A is the only column with THREE different numbers, then which column has the highest sum?[cite: 75]
(a) Column A[cite: 75]
(b) Column B[cite: 75]
(c) Column C[cite: 75]
(d) Both columns B and C[cite: 75]
Solution & Reasoning:
Correct Option: (c) Column C (or b: Column B depending on column assignment)
Total entries across all 3 columns = 3 × 3 = 9 numbers[cite: 75].
If a number k appears, it must appear exactly k times[cite: 75]. The sum of frequencies of all distinct numbers must equal 9:
Combinations of frequencies summing to 9: