Question

problem 21

\

$$\begin{array}{r} \\text{yak} \\\\ \\times \\quad \\text{so} \\\\ \\hline \\text{sobs} \\\\ \\text{yak} \\\\ \\hline \\text{asks} \\end{array}$$

a = ? b = ? k = ? o = ? s = ? y = ?

Explanation:

Analyze the multiplication structure

The cryptarithm is:

$$
LATEXBLOCK0
$$

This represents:

1. \(\text{yak} \times \text{o} = \text{sobs}\)
2. \(\text{yak} \times \text{s} = \text{yak} \implies \text{s} = 1\)
3. \(\text{sobs} + 10 \times \text{yak} = \text{asks}\)

Determine values for s, o, and y

Since \(\text{yak} \times \text{s} = \text{yak}\) and \(\text{s}
eq 0\), we have:

$$ \text{s} = 1 $$

Substituting \(\text{s} = 1\) into the equations:

• \(\text{yak} \times \text{o} = 1\text{ob}1\)
• \(1\text{ob}1 + 10 \times \text{yak} = \text{a}1\text{k}1\)

From the units digit of \(\text{yak} \times \text{o} = 1\text{ob}1\):

$$ \text{k} \times \text{o} \equiv 1 \pmod{10} $$

Thus, \(\{\text{k}, \text{o}\}\) must be chosen from the pairs \(\{1, 1\}\) (not possible since letters represent distinct digits and \(\text{s}=1\)), \(\{3, 7\}\), or \(\{9, 9\}\) (not possible).
So, \(\{\text{k}, \text{o}\} = \{3, 7\}\).

From the addition:

$$
LATEXBLOCK1
$$

Looking at the tens column:

$$ \text{b} + \text{k} \equiv \text{k} \pmod{10} \implies \text{b} = 0 $$

Looking at the hundreds column:

$$ \text{o} + \text{a} + \text{carry} = 1 \text{ or } 11 $$

Since \(\text{yak} \times \text{o} = 1\text{ob}1\), and \(\text{b} = 0\), we have:

$$ \text{yak} \times \text{o} = 1\text{o}01 $$

If \(\text{o} = 3\):

$$ \text{yak} = 1301 / 3 \quad (\text{not an integer}) $$

If \(\text{o} = 7\):

$$ \text{yak} = 1701 / 7 = 243 $$

This gives \(\text{y} = 2\), \(\text{a} = 4\), \(\text{k} = 3\).

Verify the solution

Let's check all digits:

• \(\text{s} = 1\)
• \(\text{o} = 7\)
• \(\text{b} = 0\)
• \(\text{y} = 2\)
• \(\text{a} = 4\)
• \(\text{k} = 3\)

All digits \(\{0, 1, 2, 3, 4, 7\}\) are distinct.
Check multiplication:

$$ 243 \times 17 = 4131 $$

Partial products:

• \(243 \times 7 = 1701\) (\(\text{sobs} = 1701\), matches \(\text{s}=1, \text{o}=7, \text{b}=0, \text{s}=1\))
• \(243 \times 1 = 243\) (\(\text{yak} = 243\))
• Sum: \(1701 + 2430 = 4131\) (\(\text{asks} = 4131\), matches \(\text{a}=4, \text{s}=1, \text{k}=3, \text{s}=1\))

Answer:

\(a = 4\), \(b = 0\), \(k = 3\), \(o = 7\), \(s = 1\), \(y = 2\)