\(\begin{array}{r} D\quad A\quad N\quad G\quad O \\ \times\qquad K \quad Z\quad O\quad K \\ \hline D\quad A\quad I\quad K\quad A\quad Z\quad o\quad K\quad U \end{array}\)

**TL;DR - bolded text**

"Nagisa, look what I have got here. A dango math cryptogram!" Tomoya said, waving his smartphone excitedly at Nagisa's face.

"Um, what's a cryptogram, Tomoya?" Nagisa asked. "I have never heard about it before."

"It is a puzzle of some sort, silly. The rules are simple. **Each alphabet you see here represents a number from 0 to 9.** There are a few catches though." Tomoya explained.

"What are they?" Nagisa asked, tilting her head cutely in puzzlement. Tomoya suddenly experienced an inexplicable spike in his heart rate.

"Well, to throw in some fun, **the four 'K's in the cryptogram represent two different pairs of numbers**. Besides that, notice how **the O in DAIKAZOKU is a tad bit smaller than usual**?"

"Uh-huh," Nagisa answered.

"It is actually a mischievous baby dango! It sneaked in there while I was designing the cryptogram," Tomoya exclaimed dramatically. "**The little o is essentially a wildcard and can represent any number from 0 to 9, even a number already represented by another letter**."

"Aww, that's so cute, Tomoya-kun. But you don't have to invent a story just because the cryptogram won't work if that has been a regular O."

Large beads of sweat started precipitating on Tomoya's forehead.

*uneasy laughter*

"Well, while we are at it, let me give you some hints. **The integer represented by DANGO has only 2 prime factors, one of which is 5 and another of which consists of 4 digits. Besides that, the first four letters, D, A, I, K of DAIKAZOKU forms a strictly increasing arithmetic progression**." Tomoya said.

"Umm, that's difficult. Do I get a dango plushie if I am able to solve this?" Nagisa asked.

"You still want a dango plushie that badly? I have bought you three plushies already, remember?"

"But they are too cute to resist! How do I solve this problem? It is really difficult!"

"My, my, Nagisa. You aren't any smarter than a 8th grader, are you?" teased Tomoya.

"Hey, I don't remember learning this in my 8th grade!" Nagisa said, puffing her cheeks.

Can you help Nagisa obtain the plushie by deducing **which number the wildcard (little o) represents?**

Dango Daikazoku 団子大家族 literally translates to "A Big Dango Family"

This problem is part of the question set Mathematics in Anime

The characters (Nagisa and Tomoya) and illustrations of dango daikazoku belong to KEY Studios (creator of CLANNAD visual novel+ anime series) and original creator of the fanart respectively. I do not, in any way, own these amazing pieces of work.

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