Solve the following: (i) (ii) (iii) (iv) (v) (vi)

Understand the Question In cryptarithmetic (alphametic) puzzles, each letter represents a unique single-digit integer from to . Leading digits of multi-digit numbers cannot be . To solve each equation, express the numbers in their decimal expanded form (e.g., ) and use algebraic simplification combined with digit properties. Part (i) Solve Step 1 · Form and Solve the Equation Express numbers in decimal form: Substitute into the equation: Since is a two-digit number, : Since must be a non-zero single digit, . Substitute : Thus, and . Verification: , which matches . Answer > (i) Part (ii) Solve Step 1 · Form and Solve the Equation Express numbers in decimal form: Substitute into the equation: Since , we have . Therefore, the leading digit must be . Substitute : Since multiplying by results in a units digit of or , must be or : If : (not a single digit). If : Thus, . Verification: , which matches . Answer > (ii) Part (iii) Solve Step 1 · Analyze Column Multiplications Express numbers in decimal form: Arranging column-wise: From the hundreds place: . Since , and carry from tens place is . From the tens place: , where is the carry from : Since can be or : Case 1: If : Here, (all unique, valid digits). Case 2: If : Here, (all unique, valid digits). Answer > (iii) Part (iv) Solve Step 1 · Form and Solve the Equation Express numbers in decimal form: Since , we have . Thus, can be or . The units digit of must equal : For : (ends in ), so . For : (ends in ), so . Testing Cases: Case 1: Digits are unique and valid. Case 2: This gives a three-digit product, which is not allowed. Answer > (iv) Part (v) Solve Step 1 · Form and Solve the Equation Express numbers in decimal form: Since is a three-digit number, : Also, the leading digit of must equal : If : . For , . If : . For , . If : . For , . All other products do not satisfy uniqueness or lead to a contradiction with the leading digit . Answer > (v) Part (vi) Solve Step 1 · Form and Solve the Equation Express numbers in decimal form: Divide both sides by : Since , must be a multiple of , and must be a multiple of . Since and are non-zero single digits: Verification: , which matches . Answer > (vi) Common Mistakes Treating Digits as Multiplication: Forgetting that means , not . Non-Unique Digits: Assigning the same value to different letters (e.g., setting ). Allowing Leading Zeros: Allowing the first digit of a multi-digit number to be .

Divisibility and Multiples | IT

Question 41

Solve the following:

(i) $\text{UT} \times 3 = \text{PUT}$

(ii) $\text{AB} \times 5 = \text{BC}$

(iii) $\text{L2N} \times 2 = \text{2NP}$

(iv) $\text{XY} \times 4 = \text{ZX}$

(v) $\text{PP} \times \text{QQ} = \text{PRP}$

(vi) $\text{JK} \times 6 = \text{KKK}$

Solution

Understand the Question

In cryptarithmetic (alphametic) puzzles, each letter represents a unique single-digit integer from $0$ to $9$ .
Leading digits of multi-digit numbers cannot be $0$ .
To solve each equation, express the numbers in their decimal expanded form (e.g., $\text{AB} = 10A + B$ ) and use algebraic simplification combined with digit properties.

(i) Solve $\text{UT} \times 3 = \text{PUT}$

Step 1 · Form and Solve the Equation

Express numbers in decimal form: $\text{UT} = 10U + T$ $\text{PUT} = 100P + 10U + T$

Substitute into the equation:

\begin{aligned} (10U + T) \times 3 &= 100P + 10U + T \\[0.6em] 30U + 3T &= 100P + 10U + T \\[0.6em] 20U + 2T &= 100P \\[0.6em] 10U + T &= 50P \end{aligned}

Since $\text{UT} = 10U + T$ is a two-digit number, $10 \le 50P \le 99$ : $0.2 \le P \le 1.98$

Since $P$ must be a non-zero single digit, $P = 1$ .

Substitute $P = 1$ :

\begin{aligned} 10U + T &= 50 \times 1 \\[0.6em] 10U + T &= 50 \end{aligned}

Thus, $U = 5$ and $T = 0$ .

Verification: $50 \times 3 = 150$ , which matches $\text{UT} \times 3 = \text{PUT}$ .

Answer

(i) $\text{U} = 5, \quad \text{T} = 0, \quad \text{P} = 1$

(ii) Solve $\text{AB} \times 5 = \text{BC}$

Step 1 · Form and Solve the Equation

Express numbers in decimal form: $\text{AB} = 10A + B$ $\text{BC} = 10B + C$

Substitute into the equation:

\begin{aligned} (10A + B) \times 5 &= 10B + C \\[0.6em] 50A + 5B &= 10B + C \\[0.6em] 50A &= 5B + C \end{aligned}

Since $\text{BC} < 100$ , we have $\text{AB} < \dfrac{100}{5} = 20$ . Therefore, the leading digit must be $A = 1$ .

Substitute $A = 1$ :

\begin{aligned} 50 \times 1 &= 5B + C \\[0.6em] 50 &= 5B + C \end{aligned}

Since multiplying by $5$ results in a units digit of $0$ or $5$ , $C$ must be $0$ or $5$ :

If $C = 0$ : $5B = 50 \implies B = 10$ (not a single digit).
If $C = 5$ :

\begin{aligned} 50 &= 5B + 5 \\[0.6em] 5B &= 45 \\[0.6em] B &= 9 \end{aligned}

Thus, $A = 1, B = 9, C = 5$ .

Verification: $19 \times 5 = 95$ , which matches $\text{AB} \times 5 = \text{BC}$ .

Answer

(ii) $\text{A} = 1, \quad \text{B} = 9, \quad \text{C} = 5$

(iii) Solve $\text{L2N} \times 2 = \text{2NP}$

Step 1 · Analyze Column Multiplications

Express numbers in decimal form:

\begin{aligned} (100L + 20 + N) \times 2 &= 200 + 10N + P \\[0.6em] 200L + 40 + 2N &= 200 + 10N + P \end{aligned}

Arranging column-wise:

\begin{array}{r} \text{L 2 N} \\ \times \quad 2 \\ \hline \text{2 N P} \end{array}

From the hundreds place: $2L + \text{carry} = 2$ . Since $L \ne 0$ , $L = 1$ and carry from tens place is $0$ .

From the tens place: $2 \times 2 + c_1 = N$ , where $c_1$ is the carry from $N \times 2$ : $4 + c_1 = N$

Since $c_1$ can be $0$ or $1$ :

Case 1: If $c_1 = 0$ :

\begin{aligned} N &= 4 + 0 = 4 \\[0.6em] N \times 2 &= 4 \times 2 = 8 \implies P = 8 \end{aligned}

Here, $L = 1, N = 4, P = 8$ (all unique, valid digits).

Case 2: If $c_1 = 1$ :

\begin{aligned} N &= 4 + 1 = 5 \\[0.6em] N \times 2 &= 5 \times 2 = 10 \implies P = 0 \end{aligned}

Here, $L = 1, N = 5, P = 0$ (all unique, valid digits).

Answer

(iii) $\text{L} = 1, \, \text{N} = 4, \, \text{P} = 8 \quad \text{or} \quad \text{L} = 1, \, \text{N} = 5, \, \text{P} = 0$

(iv) Solve $\text{XY} \times 4 = \text{ZX}$

Step 1 · Form and Solve the Equation

Express numbers in decimal form:

\begin{aligned} (10X + Y) \times 4 &= 10Z + X \\[0.6em] 40X + 4Y &= 10Z + X \\[0.6em] 39X + 4Y &= 10Z \end{aligned}

Since $\text{ZX} < 100$ , we have $\text{XY} < \dfrac{100}{4} = 25$ . Thus, $X$ can be $1$ or $2$ .

The units digit of $4Y$ must equal $X$ :

For $Y = 3$ : $4 \times 3 = 12$ (ends in $2$ ), so $X = 2$ .
For $Y = 8$ : $4 \times 8 = 32$ (ends in $2$ ), so $X = 2$ .

Testing Cases:

Case 1: $X = 2, Y = 3$ $23 \times 4 = 92 \implies Z = 9$ Digits $(2, 3, 9)$ are unique and valid.
Case 2: $X = 2, Y = 8$ $28 \times 4 = 112$ This gives a three-digit product, which is not allowed.

Answer

(iv) $\text{X} = 2, \quad \text{Y} = 3, \quad \text{Z} = 9$

(v) Solve $\text{PP} \times \text{QQ} = \text{PRP}$

Step 1 · Form and Solve the Equation

Express numbers in decimal form: $\text{PP} = 11P, \quad \text{QQ} = 11Q, \quad \text{PRP} = 101P + 10R$

\begin{aligned} (11P) \times (11Q) &= 101P + 10R \\[0.6em] 121PQ &= 101P + 10R \end{aligned}

Since $\text{PRP}$ is a three-digit number, $100 \le 121PQ \le 999$ : $0.82 \le PQ \le 8.25$

Also, the leading digit of $121PQ$ must equal $P$ :

If $PQ = 2$ : $121 \times 2 = 242 \implies P = 2, R = 4$ . For $P = 2$ , $Q = 1$ .
If $PQ = 3$ : $121 \times 3 = 363 \implies P = 3, R = 6$ . For $P = 3$ , $Q = 1$ .
If $PQ = 4$ : $121 \times 4 = 484 \implies P = 4, R = 8$ . For $P = 4$ , $Q = 1$ .

All other products $PQ$ do not satisfy uniqueness or lead to a contradiction with the leading digit $P$ .

Answer

(v) $\text{P} = 2, \, \text{Q} = 1, \, \text{R} = 4 \quad \text{or} \quad \text{P} = 3, \, \text{Q} = 1, \, \text{R} = 6 \quad \text{or} \quad \text{P} = 4, \, \text{Q} = 1, \, \text{R} = 8$

(vi) Solve $\text{JK} \times 6 = \text{KKK}$

Step 1 · Form and Solve the Equation

Express numbers in decimal form: $\text{JK} = 10J + K, \quad \text{KKK} = 111K$

\begin{aligned} (10J + K) \times 6 &= 111K \\[0.6em] 60J + 6K &= 111K \\[0.6em] 60J &= 105K \end{aligned}

Divide both sides by $15$ : $4J = 7K$

Since $\gcd(4, 7) = 1$ , $J$ must be a multiple of $7$ , and $K$ must be a multiple of $4$ . Since $J$ and $K$ are non-zero single digits:

$J = 7$
$K = 4$

Verification: $74 \times 6 = 444$ , which matches $\text{JK} \times 6 = \text{KKK}$ .

Answer

(vi) $\text{J} = 7, \quad \text{K} = 4$

Common Mistakes

Treating Digits as Multiplication: Forgetting that $\text{AB}$ means $10A + B$ , not $A \times B$ .
Non-Unique Digits: Assigning the same value to different letters (e.g., setting $P = Q = 2$ ).
Allowing Leading Zeros: Allowing the first digit of a multi-digit number to be $0$ .

More on Divisibility Shortcuts

Divisibility Shortcuts for Other Numbers

How can we find out if a number is divisible by 6?

Q31

Will checking its divisibility by its factors 2 and 3 work? Use the shortcuts for 2 and 3 on these numbers and divide each number by 6 to verify— 38, 225, 186, 64.

Q32

How about checking divisibility by 24? Will checking the divisibility by its factors, 4 and 6, work? Why or why not?

Q33

What property do you think this digital root will have? Recall that we did this while finding the divisibility shortcut for 9.

Q34

Between the numbers 600 and 700, which numbers have the digital root: (i) 5, (ii) 7, (iii) 3?

Q35

Write the digital roots of any 12 consecutive numbers. What do you observe?

Q36

Now, find the digital roots of some consecutive multiples of (i) 3, (ii) 4, and (iii) 6.

Q37

What are the digital roots of numbers that are 1 more than a multiple of 6? What do you notice?

Try to explain the patterns noticed.

Q38

I’m made of digits, each tiniest and odd, No shared ground with root #1—how odd!

My digits count, their sum, my root— All point to one bold number’s pursuit— The largest odd single-digit I proudly claim.

What’s my number? What’s my name?

Q39

Solve the cryptarithms given below:

Q40

Try this now: $\text{GH} \times \text{H} = 9\text{K}$ .

This means a 2-digit number multiplied by a 1-digit number gives another 2-digit number in the 90s. Observe the letters corresponding to the units digits in this cryptarithm. Pick the solution to this question from the options given below: $11 \times 9 = 99$ , $12 \times 8 = 96$ , $46 \times 2 = 92$ , $24 \times 4 = 96$ , $47 \times 2 = 94$ , $31 \times 3 = 93$ , $16 \times 6 = 96$ .

Q41

Solve the following:

(i) $\text{UT} \times 3 = \text{PUT}$

(ii) $\text{AB} \times 5 = \text{BC}$

(iii) $\text{L2N} \times 2 = \text{2NP}$

(iv) $\text{XY} \times 4 = \text{ZX}$

(v) $\text{PP} \times \text{QQ} = \text{PRP}$

(vi) $\text{JK} \times 6 = \text{KKK}$

Question 41

More questions in IT

Breaking Even

Pairs to Make Fours

Always, Sometimes, or Never

More on Divisibility Shortcuts

Divisibility Shortcuts for Other Numbers